Money Capital in the Theory of the Firm: A Preliminary Analysis

Money capital in the theory of the firm

Money capital in the theory of the firm A preliminary analysis

DOUGLAS VICKERS University of Massachusetts

The right of the University of Cambridge to print and sell all manner of books was granted by Henry VIII in 1534. The University has printed and published continuously since 1584.

CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1987 First published 1987 Printed in the United States of America Library of Congress Cataloging-in-Publication Data Vickers, Douglas, 1924Money capital in the theory of the firm. Bibliography: p. 1. Capital. 2. Business enterprises. I. Title. HB501.V515 987 658.T52 86-23232 British Library Cataloguing in Publication Data Vickers, Douglas Money capital in the theory of the firm : a preliminary analysis. 1. Working capital I. Title 658.1'5244 HG4028.W65 ISBN 0 521 32841 1

Contents

Preface

page ix

PART I: Theoretical issues and analytic motivation 1 The firm in a monetary economy Time 1 Real-time choice-decision point 2 Production time period 3 Real capital asset investment 4 Liquid asset portfolio 5 Debt capital financing 6 Residual ownership investment in the Uncertainty Money Plan of the book 2

3

firm

3 4 7 9 9 10 12 13 15 18 19

Assets, capital, and capitalization Stocks and flows in economic theory Balance sheet, or economic position statement Transformation of assets and liabilities: dynamics of the balance sheet Income statement, or economic performance statement Operational and financial leverage Economic decisions in the firm: production, capital, and finance Economic decisions in the firm and economic position and performance statements Economic valuation, or capitalization of income streams

22 23 24

The concept and relevance of economic value Time value of money Present values offuture expected series of money payments Application to the firm s investment decisions Economic value and the relevance of separation theorems

41 42 43 44 50

27 28 31 34 37 38

vi

Contents

PART II: The neoclassical tradition 4

5

6

Production, pricing, investment, and financing interdependence in the firm Notation employed in the model Preliminary comments on elements of the model 1 Costs of debt and equity capital 2 Revenue function 3 Operating flow costs of factor inputs 4 Money capital requirement coefficients 5 Net working capital asset requirements 6 Money capital requirement function 7 Money capital availability constraint Initial approximation to the constrained optimization model Marginal productivity of money capital Effective cost and product contours and factor substitution Constrained objective function and economic valuation of the ownership of the firm Expansion of equity investment A further approximation to money capital optimization Probability, risk, and economic decisions Probability and probability distributions Asset combinations and covariance between asset returns Correlation between random variables Linear combinations of random variables: expected values, variances, and covariances Covariance matrix Covariance matrix of a weighted sum of random variables Variance of a weighted sum of random variables Application of probability and risk: two-asset portfolio opportunity locus Utility, uncertainty, and the theory of choice Utility function Utility and probability: investor s risk-aversion utility function Axiomatic basis of stochastic utility Construction of the utility function Example of a derived utility function Uniqueness of the utility function The firm s choice of output under selling price uncertainty

59 60 61 61 62 62 65 66 67 67 68 70 71 73 11 79 82 82 88 91 93 96 97 102 103 108 108 112 119 122 124 111 129

7

8

9

Contents Financial asset markets and the cost of money capital Assumption content of the equilibrium theory of financial asset prices Portfolio asset choice Risk-free asset opportunity Asset market equilibrium condition Nonstandard results of the equilibrium asset pricing theory A utility-theoretic interpretation Wealth allocation separation theorem The cost of money capital The cost of money capital: further analysis and controversy Illustration of the full marginal cost of relaxing the money capital availability constraint Weighted average cost of capital The question of business risk Modigliani-Miller hypotheses: weighted average cost of capital revisited Adjustment for the firm' s tax liability Entity theory of the firm s capital structure and the equilibrium theory of financial asset markets Growth of the firm and the cost of equity capital Variable growth rates and the cost of equity capital Comparison of equity capital costs The investment expenditure project Investment decision criterion Relevant cost of capital Investment project expected cash inflows Probabilistically reducible risk and investment decisions

vii 133 133 136 139 143 147 150 151 153 156 156 160 162 164 111 111 175 177 178 180 180 181 184 185

PART III: Postclassical perspectives 10 Neoclassicism and an alternative perspective Production and factor employment Output markets and competitive conditions Theory of utility Logic of constrained optimization Historic time, uncertainty, and knowledge General equilibrium postulates An alternative theory of money capital investment Uncertainty and potential surprise

191 191 193 193 194 194 195 196 196

viii

Contents

11 Production and the place of money capital Operating characteristics of the Money capital employment decision The firm's optimum-structure decisions

firm

12 Uncertainty and decisions in the firm Possibility and economic decisions Potential-surprise function Investment decision criterion Interpretation of economic value The possibility of project reversal Utility reappraised A final note on probability Further applications and the place of economic judgment References Index

197 201 205 210 212 213 215 219 225 226 227 228 230 232 239

Preface

The place of money capital in the theory of the firm is interdependent with the analysis of production, pricing, and capital investment. In this book I have examined those areas of economic theory that bear most directly on that analytical interdependence. In doing so, I have had principally in mind the needs of advanced students, including those who are making an initial approach to the subject at both undergraduate and graduate levels. I hope that my professional colleagues will be interested eavesdroppers, and that in those areas in which some novelty is proposed it might be thought that prospects exist for a meaningful reconstruction and new advances in the theory of the firm. I have not set out, however, to construct a theory of the firm in its entirety. I have addressed the main issues of the relation between the firm's employment of real capital and money capital, and I have examined the linkage of those questions to the cost and availability of finance and relevant decision making in the firm. My perception of the needs of the student has determined the method and level of exposition. I have found, for example, that students have benefited from a reasonably painstaking exposition of the statistical foundations of probability theory and from a good degree of motivation for the mathematical development of such advanced topics as the utility function defined over stochastic arguments, the equilibrium theory of financial asset prices and yields, the cost of money capital, and investment decision criteria. A large number of more advanced treatments of these issues are referred to in the course of the argument and are noted in the list of references, but they are not all developed to the degree that might have been desirable if different objectives had been in view. In the neoclassical tradition, the theory of the firm has not provided a rigorous working out of the analytical interrelations I have in view. As Boulding has lamented, the firm "as generally presented in the textbooks . . . is a strange bloodless creature." It is usually examined without any recognition of the place and significance of money capital. No serious attention has generally been given to sorting out the conceptual distinctions between, and the mutual dependence of, real capital and money capital. A lack of clarity exists as to the meaning and interpretation of capital as a factor of production whose cost may or may not be coordinate with that of factors of production in general. The need exists to reconsider the timeless, static, certainty or certainty ix

x

Preface

equivalent analysis of the firm to which our absorption with the general equilibrium apparatus has directed us. The manner in which that need is addressed is indicated in more detail in the outline of the plan of the book in Chapter 1, which makes extensive prefatory observations unnecessary. After a preliminary sorting out of basic ideas and areas of analysis in the three chapters of Part I, I have arranged the main body of the work under the headings of "The neoclassical tradition" in Part II and "Postclassical perspectives" in Part III. The six chapters of Part II refurbish and expand the neoclassical theory of the firm in such a way as to approach, from that perspective, the possibility of an analytical integration of the kind I have referred to. In Part III the argument is relieved of the weight of the conventional neoclassical assumptions. The chapters of that final part present an alternative view of the firm's pricing, production, investment, and financing interdependence in which, however, certain aspects of the preceding analysis are seen in a different guise and are accorded a different analytical significance. In the final chapter a new analysis is given of the firm's decision problems in the context of the uncertainties and ignorance in which economic action is bound. The neoclassical apparatus of the probability calculus and stochastic utility of Part II is replaced by potential surprise and a new Decision Index in Part III. Parts of the book are unavoidably influenced by my previous writing on this subject, and that is referred to where appropriate. I am grateful to the University of Pennsylvania Press for permission to reproduce in amended form part of the argument contained in my Financial Markets in the Capitalist Process. My heavy indebtedness to the architects of the neoclassical theory will be clear from the chapters of Part II, and I have endeavored to make full acknowledgment to all those scholars whose work has formed my impressions and understanding in these areas. I am indebted also to those who have preceded me in the task of erecting a new and hopefully more satisfying and relevant body of theory, and I trust again that my acknowledgment of that debt will be clear in the argument of the book. My thanks are due also to my colleagues, Professors Donald W. Katzner and Randall Bausor, with whom I have discussed many of the ideas contained in the book, and who gave me the benefit of their critical evaluation at earlier stages of the work. I absolve them, however, from responsibility for the blemishes that remain. I happily record my deepest thanks to Ann Hopkins, who undertook responsibility for the word processing and the production of the manuscript.

PART I

Theoretical issues and analytic motivation

CHAPTER 1

The firm in a monetary economy

Time, uncertainty, and money form an analytical triad that economic theory, if it aspires to realism and relevance, must take seriously into account. The historic, unidirectional flow of time carries with it the inescapable reality of uncertainty and the ignorance in which we are bound. Our analytical constructions that aim to explain the world must confront the influence, the ineluctabilities, with which the passing of time presses on our experience and understanding. Knowledge, it has been said, cannot be gained before its time (Lachmann, 1959, p. 73). Alfred Marshall, the architect of English neoclassical economics, cautioned that "we cannot foresee the future perfectly. The unexpected may happen" (1920, p. 347), and he pointed to the difficulties that arise, as a result, for economic decisions and action. Keynes's observation, when contemplating the impact of the future on economic behavior, that 44 we simply do not know" (1937, p. 185), recalls his well-known indictment of the classical economics and its attempt to evade the future by a probabilistic reductionism. Many issues in the theory of the firm are brought into focus by these considerations. Terence Hutchison, whose work has provided luminous perspectives on economic thought, has seen these issues laced together in their interdependence. Noting that "uncertainty is present . . . in principle . . . with any piece of conduct in this world" (1960, p. 86), Hutchison observed with reference to the classical analysis based on its "Fundamental Assumption" of maximization that "the only way to make sense of most formulations of the Fundamental Assumption is to add the assumption of 'perfect expectations' " (p. 105). But perfect expectations, we shall argue at length, evade the real questions that claim our analytical attention. If, in fact, uncertainty in economics could be escaped by allowing the assumption of perfect expectations to abolish the future, we would abolish also the third element of our analytical triad. There would in that case be no need for money. For there would then be no function for money to perform. Money, we shall see, is a time-and-uncertainty phenomenon. "An analysis of a world with any uncertainty in it," Hutchison has argued, "and particularly an analysis which takes into account the factor of money (which can be construed as a sign that uncertainty is present . . .), cannot start from the same assumption of 'sensible' or 'rational' conduct as that applicable in a world without uncer3

4

I Theoretical issues and analytic motivation

tainty, with which, consciously and explicitly or not, the bulk of pure economic theory from Ricardo onwards appears to have been concerned" (1960, p. 88). Moreover, "the assumption of a tendency towards equilibrium implies, on the usual definition, the assumption of a tendency towards perfect expectations . . . and the disappearance of money" (1960, p. 107). But the issues of time, uncertainty, and money have not adequately informed the theory of the firm in its received traditions. Notwithstanding the achievements of the Robinsonian-Chamberlinian revolution of the 1930s, the theory of the firm quickly accommodated to the timeless, static, competitive assumptions of the Walrasian equilibrium analysis (Robinson, 1969; Chamberlin, 1933).* A slightly fuller, but at this stage intentionally incomplete, consideration of some aspects of these questions will provide a basis for the argument of the following chapters. Time The theoretical problem of time is highlighted by the briefest look at the history of our subject. Marshall had insisted, at the high tide of neoclassicism, on "the great importance of the element of time . . . the source of many of the greatest difficulties in economics" (1920, pp. 347, 109). But his theory of the representative, or "average," firm, though it was introduced in a context that took account of the evolution and decline of actual firms, served, as did Pigou's notion of the optimum or equilibrium size of the firm, as an intellectual construct designed to accommodate the realities of historic time to what was to become a timeless and static theory (see Robinson, 1969, pp. v-vi). Marshall had, of course, proposed the notion of economic equilibrium as analogous to "the mechanical equilibrium of a stone hanging by an elastic string, or of a number of balls resting against one another in a basin" (1920, p. 323), and he spoke of "equilibrium price" and "equilibrium amount" as these described possible market outcomes (p. 345). He went a good distance in accommodating his argument to the reality of the actual time span in which economic events transpired, and his conception of the short-run and the longrun period has become a familiar part of the analytical economist's tool kit (1920, p. 369f.). But Marshall hoped that all of his arguments about equilibrium, along with his use of biological and mechanical analogies and "all suggestions as to economic rest," could be seen as "merely provisional, used only to illustrate 1

Advances beyond the earlier static equilibrium theory, in the direction of intertemporal analyses, sequence models, and temporary equilibrium, can be inspected in Weintraub (1979). Weintraub observes, however, that while "the path through [Walrasian] disequilibrium theory requires one to step through analytic time . . . 'real time' adjustment is badly handled in all these models" (pp. 125, 127).

1 The firm in a monetary economy

5

particular steps in the argument, and to be thrown aside when that is done" (1920, p. 366). His cautions, however, were substantially ignored. The analogies absorbed the substance in the main body of economic analysis. The Marshallian concern for the real and actual passing of time was transmuted in the 1930s in the manner of Joan Robinson's influential Economics of Imperfect Competition, where she stated, in elevating the equilibrium theoretic tradition, that "the technique set out in this book is a technique for studying equilibrium positions. No reference is made to the effects of the passage of time" (1969, p. 16). She did refer at a later time, in an admirable passage in the second edition of her book, to her "shameless fudge" in having made an "analysis which in reality consists of comparisons of static equilibrium positions . . . dressed up to appear to represent a process going on through time" (p. vi). Robinson's pathbreaking work in the theory of the firm, however, still informs the traditional textbook treatments of the subject. She had directed attention away from the pervasive notion of perfect competition in economic analysis to the notion of monopoly, or to the concept of the economic uniqueness of the firm. Commenting on the catalytic significance of Sraffa's famous article of 1926 (Sraffa, 1926) and dissatisfied with the confusion in theory stemming from "the logical priority of perfect competition," Robinson observed that "no sooner had Mr. Sraffa released the analysis of monopoly from its uncomfortable pen in a chapter in the middle of the book than it immediately swallowed up the competitive analysis without the smallest effort. The whole scheme of analysis, composed of just the same elements as before, could now be arranged in a perfectly uniform manner, with no awkward cleavage in the middle of the book" (Robinson, 1969, pp. 3-4). John Hicks, whose Value and Capital in 1938 (Hicks, 1946) had substantially awakened English economics to the Walrasian general equilibrium theory, has recently reflected on this line of theoretical development by observing: "Why is it that the theory of monopolistic competition, or imperfect competition, to which so much attention was paid in the thirties, now looks so faded? Because it is quite shockingly out of time" (1976, p. 149, italics in original). Walras himself had recognized that time had to come into the picture. Consider the manner in which he specified his models of economic exchange and production. "In exchange/' he said, "commodities do not undergo any change. When a price is cried, and the effective demand and offer corresponding to this price are not equal, another price is cried for which there is another corresponding demand and offer. In production, productive services are transformed into products. After certain prices for services have been cried and certain quantities of products have been manufactured, if those prices and quantities are not the equilibrium prices and quantities, it will be necessary not only to cry new prices but also to manufacture revised quantities of prod-

6

I Theoretical issues and analytic motivation

ucts" (1953, p. 242, italics added). In the italicized clauses, the realization of a process and a lapse of time emerges. "Production," Walras acknowledged, "requires a certain lapse of time." But the "complication" is immediately assumed away. "We shall resolve the . . . difficulty purely and simply by ignoring the time element at this point" (1953, p. 242). For Walras, there could be, in the economics of exchange, no false trading, or the consummation of transactions at other than equilibrium prices (see Hicks, 1946, p. 128), and in the production model there could be no false production. All bids and offers on all markets, and all tentative decisions, were understood to be notional so long as the search for the equilibrium price and quantity vectors continued, and transactions were effected only at the finally announced equilibrium prices (see Vickers, 1978, p. 14f.). This structure of thought, from Robinson and Chamberlin through Hicks and the neo-Walrasians, has continued to influence the theory of the firm. It has failed to distinguish between what can be referred to as logical or analytic and real historic time. When the analysis has departed from the earlier assumptions of perfect expectations or certainty or certainty-equivalents, the future has generally been collapsed to the present by probability reduction methods. It has been imagined that although the future is unknown (and unknowable), nevertheless it is possible to assume, for decision purposes, that the future-dated variables in which we are interested can be described by subjectively assigned probability distributions and that the expected values of those variables can be unambiguously discounted to the present. By these methods, both uncertainty and the future are effectively abolished. True residual uncertainties have been metamorphosed to probabilistically reducible risks. We live, it is supposed, in risky conditions. But we know, or can assume that we know, the forms of the probability distributions that describe the possibilities ahead of us. In that strong assumption, we have effectively abolished ignorance. For we know, if that is the case, the general shape of things to come and we are no longer able to be surprised. Progress is possible, we shall argue, if we recapture the sense of history and of historic time that gave credence to the earlier Marshallian analysis, and which has informed the work of Knight (1933), Keynes (1937), Robinson (1974), Shackle (1969, 1972, 1974, 1983), Hutchison (1937, 1978), Loasby (1976), Davidson (1978), Bausor (1982, 1984), Vickers (1981, 1983), and others. The relevance of time for the theory of the firm calls for analysis for several reasons: 1.

The decision maker himself is locked in the process of actual time, with implications for his knowledge possibilities and his economic status and decision potential.

2.

3. 4.

5. 6.

1

1 Thefirmin a monetary economy 7 Production in the firm takes time, and cash outflows for the purchase of factor services occur before the completion of the firm's output and the inflow of sales revenues. The firm's investment that structures the production process includes durable assets whose economic lives extend over more than a single operating time period. The firm's investment in liquid assets, notably cash and marketable securities, is influenced by the intertemporal price of money or the rate of interest, as well as by the need to provide a refuge from the pressures of uncertainty and ignorance that real time involves. As a means of raising money capital, the firm may borrow in the debt capital sector of the money capital market, and intertemporal valuations determine the cost and availability of such funds. The firm's residual owners, the holders of its equity capital, receive its residual income after the payment of all costs of operation and interest on debt capital, and as the residual risk bearers they are vitally concerned with the intertemporal prospects of the firm and its income-generating ability overtime (see Vickers, 1977, 1978, 1981, 1983, 1984, 1985b). Real-time choice-decision point

To escape from the timelessness of earlier analysis, the breakthrough to a new logical construction must be made at one specific point. The significance of historic time enters economic analysis because the actual flow of it, and the unknowable expanse of it spread out ahead of us, impinge on the taking of decisions and the making of real-world choices. Historic time is significant because of the way in which it, or more precisely our imaginative perception of the possibilities inherent in it, determines what we do in our choicedecision moments and because of the way in which the passing of time qualifies our stance at successive decision points. The individual at his decision points in time does not choose between what exists or between probability distributions of what will exist, as though future possible outcomes are determined by a random generating device that churns out the results of replicable acts and experiments. Rather, choice creates history. Choice is between acts that hold out before them skeins of possible outcomes constructed in the imagination of the person choosing, skeins of imagined outcomes that are constrained to what the individual recognizes as possible. Expectations are thus subjective in the sense that they are highly personal and individual imaginative constructions, and in a given situation one individual will construct imagined possibilities of outcomes that do not occur, and could never have

8

I Theoretical issues and analytic motivation

occurred, to other individuals (see Shackle, 1969, 1979; Vickers, 1986; Littlechild, 1979; O'Driscoll and Rizzo, 1985). In his decision moment, the individual is perforce ignorant of the future, but he is able to conceive of "imagined possible outcomes" and assign to each of them a degree of "potential surprise." We refer to potential surprise because the decision maker can assign to possible future outcomes the degree of surprise he imagines now that he would experience at a future date if a contemplated event were to occur. He does this, in ways we shall consider more fully, because the decision he confronts is, in the general case, a "unique" decision. It is unique in the sense important for economics that the making of it precludes forever the possibility of its being made, or even contemplatable, again. In many areas of economic choice, decisions are what in this sense we call "self-destructive" decisions. The decision to increase the amount of real capital employed in a firm, for example, along with the raising of money capital to finance that investment, must be regarded as a unique, nonreplicable decision. For the taking of it forever changes the firm from what it would have been if the decision to invest had not been made. Similarly, the annual rate of return on a firm's common stock cannot properly be considered a random variable that can be described by an assigned probability distribution. The firm that generated that rate of return this year was not the same firm, in many economic respects, that it was last year, or two years ago, or five years ago. Firms change their operating and financial structures, their product mix and input use, their market posture and penetration, and their technological orientation. Firms change, grow, decline, and die. The uniqueness of the decision maker's stance at his decision point in historic time, the uniqueness of his inheritance of endowment and environmental structures, and the uniqueness of his knowledge and epistemic status converge to determine the value he places on the actions he conceives to be possible and the choices that, as a result, he makes. In the flow of time, knowledge is acquired by the decision maker. That knowledge cannot be unlearned in the sense that the individual can revert, after the lapse of time, to what he was and the position he was in, in every relevant epistemological sense, before. In their unfolding in time, successive decisions are, in their character and potential, unique, since situations, knowledge, and imagined possibilities change. We are therefore concerned with "an economics of movement and change, not in the sense of a mathematical dynamic system, in which time itself has been reduced to a serially dated variable, or in which equilibrium dynamic paths may simply have replaced equilibrium states without any basic reconstruction of the thought forms employed. Rather, I am interested in change in the sense of the next steps that, in more or less well understood situations, individuals might take to their best advantage" (Vickers, 1978, p. 21).

1 The firm in a monetary economy 2

9

Production time period

The fact that production takes time raises the problem of financing the production process for the period between the hiring of factor inputs and the sale of their product output. The firm accordingly faces a cash flow problem, and this implies the need to obtain money capital in the required amounts and at a satisfactory cost. The firm's production process and product mix, along with its policies on marketing and market penetration, generate a demand for money. Its optimum holding of cash depends on the rate of interest or, as we have referred to it, the intertemporal valuation of money. This is so because the debt incurred in raising the money capital to finance the firm's asset investment will have to be repaid at a later time, and the rate of interest associated with it will depend on the spectrum of money market opportunity costs of making that money capital available. We shall keep in mind, however, not simply, or even mainly, the firm's demand for money as such. In order to sustain its operations at any desired level and structure, the firm will need to maintain an asset mix that is itself, in some sense, optimal. Among its assets, the amount that is held as money, or the proportion of the money capital available to the firm that is invested in money balances, will depend on a number of complex considerations related to the optimal use of the money capital market. This in turn will determine the structure of the liabilities reported on the firm's balance sheet. For liabilities are employed to finance the acquisition of assets. The intertemporal costs we have referred to can therefore be interpreted as the costs of raising, or varying under designated circumstances, the liabilities to which the firm has access. The costs of money capital will partly determine, also, the structure of production and marketing processes that the firm undertakes, the timing of its input and output, and the degree of real capital intensity it decides upon. 3

Real capital asset investment

The firm's decision regarding the real capital intensity of its production process implies the acquisition of fixed capital assets. In technical economic terms, the firm's production function will be considered a flow-flow function, meaning thereby that the flow of attainable output depends functionally on the flow of factor inputs. The input to the production function is not the stock of capital assets actually held by the firm but the flow of services per period of time that those assets provide. If, for example, it was technologically necessary or desirable for the firm to employ, during a designated time period, a specified number of machine tool hours of a certain technological specification, that flow of machine tool hours would properly be regarded as the input flow of factor services. At the same time, an asset would appear on the firm's balance

10 I Theoretical issues and analytic motivation sheet designated as the machine tool that provided the flow of services. But in the flow-flow production function, the machine tool is not the factor of production. If it were so regarded, we should be working not with a flowflow production function but with a stock-flow function. The question of time enters this capital usage problem in a number of ways. First, the durability of real capital assets again gives rise to the need to acquire money capital to finance the asset investment. Considerations of money capital sources and the distribution of available money capital over different possible asset mixes again come into view. Second, it may be desirable, under different possible conditions as to technology and the markets for real capital assets, to consider the optimum lives of the assets actually being used. The asset investment decision may confront a trade-off between an asset with a relatively short economically useful life that did not call for a high periodic maintenance expenditure to keep it in efficient operation and another asset, of comparable technological capacity, that had a longer economic life but required a larger periodic maintenance and servicing expenditure. The capital asset investment decision depends critically on the level and stability of the future cash inflows that the asset is expected to generate and on the present discounted value of those cash flows. That present discounted value, or present capitalized value, will at times be referred to as the economic value of the asset. A relevant investment decision criterion will compare that economic value with the money capital outlay necessary to acquire the asset and bring it to operating or income-generating condition in the firm. Involved, therefore, is a discount factor (or rate of interest or cost of money capital) at which future possible cash flow magnitudes are reduced to present values. Alternatively, the future cash flows that an asset is expected to generate might be analyzed to determine the implicit rate of return they would provide on the money capital invested in the asset. Then that rate of return, which will be referred to under appropriate conditions as the marginal efficiency of investment, might be compared with the rate of interest or the cost of raising the necessary money capital. In either event, the intertemporal value of money, or the opportunity cost of money capital as determined by the complex of money capital market conditions, and the real-time dimension of the capital usage problem come prominently into view. 4

Liquid asset portfolio

An investment in money balances is required by the firm in order to enable it to pay flow costs of production and other maturing liabilities if the timing of cash inflows from the sale of products does not mesh precisely with the demands for cash outflows (see Davidson, 1965; Baumol, 1952; Miller and Orr, 1966; Brealey and Myers, 1984, p. 677). The demand for money is, however,

1 Thefirmin a monetary economy 11 a demand for a non-income-earning asset. The effective cost of investing in it must be interpreted as an opportunity cost measured by the income sacrificed by not allocating the firm's investable money capital to alternative asset forms that offer a comparable degree of risk. If, for example, an asset existed, such as a short-term government security, that promised a high degree of marketability and liquidity, it would conceivably make sense for the firm to hold a portion, probably a significant proportion, of its liquid asset requirements in such a form. Investing liquidity in such income-earning assets, however, does involve a degree of risk, and a trade-off exists between expected rates of return on such assets and the risks they incur. In the case of short-term marketable securities, the so-called market or interest rate risk refers to the possibility that a rise in the general level of interest rates may occur during the time for which the asset is held, causing a decline in the asset's market value. In such an event, the holder of the asset will have incurred a capital loss. The firm might therefore be advised to hold liquidity in a diversified portfolio of marketable assets and not only, or even mainly, in cash (see the seminal paper, Tobin, 1958). The risks involved in such portfolio decisions have generally been assessed in terms of the dispersion of a subjectively assigned probability distribution of possible rates of return. This gives rise to what has become widely referred to as the risk-return trade-off in asset portfolio construction. As a result of recent developments in the banking and financial sector, a wider range of income-earning liquid assets has become available. Firms are able to invest temporarily surplus cash in bank certificates of deposits, which may, under certain arrangements, be negotiable or saleable in the money market, thereby permitting access to cash funds at any time. Additionally, banks and other financial institutions are now permitted to pay interest rates on business firm deposits, and such rates, following the deregulation legislation of the early 1980s, are not subject to regulatory ceilings (see Vickers, 1985a). The uncertainties inherent in the flow of time make the holding of liquid transactions balances necessary, and cash balances provide a cushion against unforeseen and unfavorable developments that would otherwise cause financial embarrassment or loss. The holding of money also provides a refuge from the pressures of uncertainty and ignorance that inhibit real economic activity. In this respect, it has potentially significant implications for the employment of real resources in the firm and in the economy. As Keynes has observed, "our desire to hold money as a store of wealth is a barometer of the degree of our distrust of our own calculations and conventions concerning the future. . . . The possession of actual money lulls our disquietude" (1937, p. 187). Money may be held when the uncertainties surrounding economic prospects make it desirable to defer the commitment of resources to real investment and the pursuit of real economic activities. To the extent that this is so, available

12

I Theoretical issues and analytic motivation

real resources will not be utilized as fully as would otherwise be possible. The firm and the economy are in that case operating within, rather than on the boundary of, attainable production opportunity sets. In this sense, the firm's holding of money and liquidity is defensive. But it may also be offensive, in the sense that the holding of money imparts a degree of flexibility to the firm's production and factor use decisions, and may permit it to take advantage of previously uncontemplated investment opportunities. 5

Debt capital financing

The possibility of the firm's use of borrowed funds and the prospects of profit on the one hand and the risks and dangers of losses due to excessive indebtedness on the other have given rise to a distinguished literature in economic theory (see Kalecki, 1937; Copeland and Weston, 1983; Minsky, 1975). The economic significance of the firm's use of debt capital is due largely to the contractual nature of the arrangements entered into in connection with it. Loans are obtained by the firm from the debt capital sector of the money capital market, from both financial intermediaries and individual investors who purchase the debt securities as a means of allocating their savings funds. Certain kinds of loans, such as short-term commercial paper issued by corporations with undoubtedly high credit ratings and some short-term loans from financial institutions, may be unsecured. But in the general case, and certainly in the case of long-term corporate debt, the loans will be contractually secured. They may be secured against certain specific assets of the firm or by specifying the order of ranking of their claims against the general income-generating ability of the firm. In exchange for money capital, the firm will issue debt certificates that specify (i) the length of time between the date of issue and the maturity date, or the date in the future on which the amount borrowed and described in the certificate will be repaid to the lender, and (ii) the rate of interest (stated on the face of the debt certificate and referred to as the "coupon rate") that the firm undertakes to pay each year on the amount of the loan. Additionally, the contract entered into between the borrowing firm and the trustees of the debt (or bond) issue will specify the nature of the rights of the debt holders in the event of the insolvency or dissolution of the firm. The debt holders, then, have what is referred to as a prior claim against the annual income and the assets of the firm. This means that the firm must pay the interest on the debt capital out of whatever income remains after paying operating costs, before any residual income can be paid to the common stockholders in the form of dividends. Moreover, in the event of the dissolution of the firm, the debt holders will have a claim against the liquidation value of the firm's assets before any distribution can be made to the equity holders.

1 Thefirmin a monetary economy 13 Time and risk are interwoven in this nexus of contractual obligations. The lenders of debt capital will need to evaluate the prospective incomegenerating ability of the firm over the period of time for which the debt will be outstanding. They will be concerned with the level of the firm's earnings, the possible trend in earnings, and the stability of the income stream in the face of economic fluctuations. The greater the assessed or envisioned risk in the contemplated income stream, the greater will be, in general, the rate of return the lenders will require in order to induce them to hold the debt. At the same time, the borrowing firm will make its own estimates of the likely level, trend, and stability of earnings and the proportion of its net operating income that will be absorbed by the contractual interest payment on the debt. If, as will generally be hoped, the rate of return earned on the money capital raised in the form of debt is greater than the rate of interest payable on the debt, the additional earnings will accrue to the residual owners, the common stockholders of the firm. In such an event, the stockholders are said to be realizing the benefit of favorable financial leverage. At the same time, however, the overall riskiness of the common stockholders' position may be increased by virtue of the additional fixed-cost financing sources (debt capital) employed in the firm. 6

Residual ownership investment in the firm

In the theory of the firm, a confusion and ambiguity frequently surrounds the treatment of capital as a factor of production. Most usually, the discussion of the factor combination problem considers capital as a factor coordinate in every analytical sense with, say, labor or other variable factors. Adequate attention is not always given to the "price" at which the capital factor is obtainable or to the manner in which its durability over time influences the specification of its cost. This analytical hiatus has stemmed from a failure to distinguish clearly between what we shall call real capital on the one hand and money capital on the other. Moreover, when that necessary distinction is established, a further question arises. In what sense, we can ask, is capital to be understood as a factor of production? Real capital, we have already said, is a factor of production. But we have indicated the sense in which, in a flowflow conception of the production function, it is not the actual real capital assets held by the firm that enter the production function as arguments or are regarded as factors of production. The capital factor is described by the flow of services that those capital assets provide per period of operating time. Money capital, on the other hand, is not a factor of production. Money capital functions as a constraint in that it provides the purchasing power that gives the firm control over factors of production and necessary cooperating assets. This distinction between real capital and money capital raises the question

14

I Theoretical issues and analytic motivation

of the forms, and the possibly optimum forms, in which money capital should be raised and made available to the firm. The detailed and intricate possibilities that actually exist can be summarized in a twofold classification of (i) debt capital and (ii) equity, or residual ownership, capital. The total equity capital employed in the firm derives from two sources. First, the sale of shares of common stock in the money capital market provides the initial infusion of funds to a firm and establishes it as a going concern. As the firm operates over time, it will distribute its residual income to the common stockholders, the providers of equity capital, in the form of dividends, or it may retain some or all of those earnings and use them to finance the expansion and development of the firm. In the latter case, the retained earnings nevertheless become the property of the equity owners, in the sense that their total stake in the firm is increased as a result. Many aspects of the relation between the firm and the money capital market have to do with the possible, or hoped for, increase in the market value of the shares of common stock that results from such a retention and reinvestment of earnings. An earlier and significant, but until recently a relatively neglected, literature in the theory of the firm wrestled with these highly important questions. 2 Bringing together these sources of money capital, we shall incorporate into the theory of the firm what we shall call a money capital availability constraint. This will require a careful analysis of the possible form of that constraint, the degree to which debt and equity capital are combined to provide money capital, and the manner in which the constraint may be relaxed by introducing marginal units of either debt or equity capital to the firm. At the latter point, the question of the intertemporal value of money again becomes relevant. For the introduction of money capital involves a liability on the part of the firm to repay that money capital at a specified later date if it is debt capital or to service the equity capital by paying dividends at the end of each operating period or increasing the equity owners' claim on the firm if the residual earnings are retained. In either event, relaxing the money capital availability constraint raises the question of the effective economic cost of doing so. Entering the analysis, therefore, is the concept of the "full marginal cost of relaxing the money capital availability constraint." Its magnitude will be affected by the general level of interest rates in the money capital market, or the market's opportunity cost of supplying money capital to the firm, and by the evaluation of the risks involved as they are seen by the supply side of the money capital market.3 That perception of risk will depend on the general 2

3

See Buchanan (1940) and Williams (1965). Seminal work in the theory of the growth of the firm appeared in Penrose (1959), Baumol (1959), Marris (1964), and Gordon (1962). For significant earlier discussion of the integration between the production and the capital investment problems in the firm, which does not, however, take account of the financing question, see Smith (1959, 1961). On the question of the money capital availability constraint, see the early but neglected paper by Lange (1936). See also Gabor and Pearce (1952, 1958) and Vickers (1968, 1970).

1 Thefirmin a monetary economy

15

estimate of the level, trend, and stability of the firm's future income and cash flow streams and on the existing debt-to-equity ratio, or the financing mix, in the firm's capital structure. The question of risk enters in a unique way in connection with the availability and cost of equity capital. The equity holders are the residual owners of the firm, they receive the residual income after all operating costs and debt interest have been paid, and they are, among the providers of money capital, the residual risk bearers. They bear the final brunt not only of what has been termed the variability risk, or the risk of variability in the firm's income stream, but also of the default or bankruptcy risk. The latter arises from the possibility, however large or small it may be, that economic conditions or failures of management may make it impossible for the firm to meet its monetary obligations for the payment of operating costs and the interest on debt capital. In the event of the insolvency and dissolution of the firm, the claims that the debt holders and other creditors have against the assets of the firm rank ahead of the residual claims of the equity owners. For these reasons, the cost of equity capital will in general be somewhat higher than that of debt capital. This cost differential is an important factor in the determination of the firm's optimum use of money capital or its optimum financing mix. Uncertainty Our discussion of time has carried along with it at many points the question of uncertainty. We have distinguished risk on the one hand from true or residual uncertainty on the other. This is necessary because of the nature of our decision points in time and their economic environment. In economic decisions, we stand at a "solitary moment," locked in the "now" between a "dead yesterday" and an "unborn tomorrow" (see Shackle, 1969, p. 14f.; 1972, p. xi; 1970). The future is not only unknown. It is unknowable. Our task is that of corraling its prospects into imagined forms and shapes that permit us to choose courses of action that will lead us from where we stand to what we envision as preferred positions. We have observed that economic theory has generally come to terms with this question by assuming that the variables in whose future possible magnitudes we are interested can be interpreted as random variables describable by subjectively assigned probability distributions. The significant logical, or epistemological, question is whether meaning can properly attach to such a procedure and whether, therefore, the probability calculus is genuinely and meaningfully applicable for our purposes. In his Three Essays a quarter of a century ago, Koopmans concluded that "our economic knowledge has not yet been carried to the point where it sheds light on the core problem of the economic organization of society: the problem of how to face and deal with uncertainty" (1957, p. 147, italics added). Sidney Weintraub had earlier wrestled with the fact that "uncertainty colors

16

I Theoretical issues and analytic motivation

all economic behavior" and that in the decisions of the firm "uncertainty will color the choice" in various ways (1949, pp. 339, 366). He was later to write of "the operations of an economy in which anticipations dominate conduct, in which the expectations have an elusive vagueness, an unsureness, and thus an uncertainty in decision making that escapes model builders, who graft illfitting mathematical probability concepts onto essentially unique and nonrepetitive events" (1977, p. 4, italics added). But traditions became securely established, and by means of the assumption of perfect expectations, or that of the reduction of uncertainty to certainty-equivalents by the use of probability devices, more comfortable knowledge assumptions were embedded in the theory. By this means, uncertainty was reduced to risk, and the probability reductionism that was employed prepared the future magnitudes of variables for discounting to present values. Uncertainty in its residual sense was abolished. Although decision makers do not know fully the outcomes that will result from their actions, they do, it was supposed, know the probability distributions of outcomes. But such an assignment of probability is, of course, an assumption of knowledge. Uncertainty, on the other hand, has to do essentially with ignorance and the absence of knowledge. What, we can ask, are the conditions that must be satisfied in order to render probabilistic arguments usable and epistemologically efficient? Two possibilities exist. First, the events we have in view might be able to be regarded as outcomes resulting from a stable event-generating mechanism. They are then generated by "the behavior of a system whose structure we can inspect, and from which we can deduce the relative frequency with which this answer or that will occur" (Shackle, 1969, p. 53). Or second, the events may be conceived as the outcomes of genuinely and completely replicable experiments. In the first case we are dealing with a priori probabilities and in the second with statistical probabilities. Unfortunately, the nature of decision situations in business and economic life, particularly in relation to production, investment, and financing decisions, are decidedly not the kind that permit the inspection of a stable structural system and the assignment of unarguable a priori probabilities. Nor, it would seem, is it possible to interpret probabilities in the economic context in the alternative or statistical sense. For the economic observations we make are most generally not the outcomes of repeatable experiments. They are what we referred to earlier as unique events, in the sense that the structure of forces determining them exists only once at a point on the passing continuum of time. Because that is so, the very uniqueness of economic events and decision situations precludes the treatment of relevant economic variables as distributional variables, or variables that are meaningfully describable by assigned probability distributions. John Hicks has recently cautioned us that "the prob-

1 Thefirmin a monetary economy

17

ability calculus, which is a powerful tool of discovery in the sciences, has seemed to be carrying all before it in economics also. . . . It is my belief that the relevance of these methods to economics should not be taken for granted" (1979, p. xi, Ch.VIII). In the handling of the unique decision moments that characterize economic life, "the probability calculus is useless; it does not apply. We are left to use our judgment, making sense of what has happened as best we can, in the manner of the historian . . . by all means let us plot the points on a chart, and try to explain them; but it does not help in explaining them to suppress their names. The probability calculus is no excuse for forgetfulness" (Hicks, 1979, pp. 121-22. See also Davidson, 1978, Chs. 2 and 3). The state of our subject so far as decision and choice under uncertainty is concerned is clear, however, from the recent exposition and summary of the theory by Hirshleifer and Riley, who define "the economics of uncertainty" as encompassing "decisions made under fixed probability beliefs" (1979). In the 150-item bibliography appended to their survey article, not a single reference appears to the conceptions of uncertainty we have just envisaged and to which we drew attention previously in the work of Hutchison, Knight, Shackle, Keynes, Loasby, Davidson, Bausor, and others. The conventional probability theorizing has been consolidated firmly in the financial theory of the firm. Mossin, for example, a principal contributor to the newer financial theories, has crystallized the issue, as he sees it, as to "what the theory of decision making under uncertainty is all about. . . . The theory deals with choices among probability distributions. By choosing one action, we get one probability distribution; by choosing another, we get another" (1973, p. 6, italics in original. See also Vickers, 1978). It may be rejoined that although in business and financial decision making it is not possible to invoke probability in either an a priori or a relative frequency sense, it is nevertheless valid to employ probability in a subjective sense and to work with subjectively assigned or "as if" probability distributions. This suggestion, however, appears wide of the mark, for it fails to understand the meaning of the irreducible uncertainty with which real-world economic decision making is actually confronted. For what, in that case, is the basis in reason or fact for the postulation of probabilities? In short, where, then, do the "probabilities" come from, and how is it possible to define exhaustively the set of possible outcomes to which the probability assignments are to be made? Moreover, if we acknowledge the uniqueness of economic events and the nondistributional character of the variables describing them, what meaning inheres in any attempt to take an average value of such variables, an expected value for example, or to perform a mathematical manipulation to provide, for example, a variance or other measure of dispersion of them? What meaning can be attached to the "variance" of unique mea-

18

I Theoretical issues and analytic motivation

sures of unique events or outcomes? We shall return to these questions in Chapter 12.

Money The relevance for the theory of money of the issues of time and uncertainty follows from the fact that money is essentially a time-and-uncertainty phenomenon. In the absence of uncertainty, as in the neo-Walrasian theories of general equilibrium, all future-dated values can be unambiguously reduced to present values, markets can be assumed to exist for all commodities tradeable at all future possible dates, and in that event goods effectively exchange directly for goods. As Hahn has summed it up, "the Walrasian economy . . . is essentially one of barter," and in an economy described by the traditional general equilibrium theory "money can play no essential role" (1970, 1971. See also Davidson, 1978, pp. xiii, xiv). Money, it has been said, is a link between the present and the future. It is also a link between the past and the present because we may carry into the present in money form part or all of the value of the resources we have decided to refrain from committing to active economic uses. In that notion lies the true function of money. It is a means of transporting purchasing power not only over space and distance but also over time. Money can serve as a store of value in the sense that when, in the presence of uncertainties, we are reluctant to commit our resources to investment in income-producing activities, we can take refuge in holding money. By holding money we are enabled to take refuge in deferred purchasing power. The holding of money is our speculation against our own lack of knowledge, precipitating a demand for money as "a substitute for knowledge" (Shackle, 1972, p. 216). The same phenomena and pressures of uncertainty also influence our understanding of the medium-of-exchange function of money. Shackle has recognized the point. Granted that it is convenient to have on hand a store of general purchasing power to satisfy a transactions demand for money, an amount of money that is required "because we are not sure what we want to buy." Then "it is being kept because of uncertainty, a petty rather than a momentous kind of uncertainty, if you wish, but for the purposes of theory, of the unity of theory, this characterization is important" (Shackle, 1974, p. 62). We shall examine in the following chapters the ways in which the exigencies of time and uncertainty provide congruent perspectives on the firm's demand for money balances and on its demand for, and investment of, money capital.

1 The firm in a monetary economy

19

Plan of the book

The place of money capital in the theory of the firm follows from the interdependence between the firm's production, pricing, capital investment, and financing decisions. In the neoclassical or textbook theories of the firm, that interdependence has not been accorded a significant analytical priority. The "real" and the "financial" theories of the firm have enjoyed separate and substantially unrelated developments. Our task in Part II of the book, therefore, will be an examination of the ways in which the neoclassical traditions in the theory of the firm may be expanded to take account of the interdependence we have referred to. The chapters of Part II will therefore be essentially in the neoclassical tradition. Based on fairly standard assumptions regarding the structure of the single-product firm, Chapter 4 will present a model of production, money capital investment, and financing interrelations. In doing so, it is not presenting a "new" theory of the firm in a fundamental sense. Rather, it attempts to demonstrate the extent to which the received traditions may be employed to the best possible advantage to throw light on the money capital problem. An alternative construction, aligned with viewpoints that we shall term postclassical rather than neoclassical, will be developed in Part III. The neoclassical tradition requires an understanding of the manner in which the probability calculus has been extensively employed to address the question of uncertainty or to abolish true uncertainty and replace it by probabilistically reducible risk. In Chapter 5 the foundations of probability analysis are laid, and the subject is developed in a self-contained fashion that is adequate for our present purposes. The foundations are thereby laid for the application of probability thought forms to financial asset choices and investment decisions and, in particular, to the question of the selection of optimum financial asset portfolios. The latter problem lies, in the neoclassical scheme of things, at the basis of the theory of the firm's cost of money capital. Chapter 6 takes up the question of utility theory and the manner in which this underlies the neoclassical theory of choice. In conditions of risk, however, the theory of utility needs to be expanded and reconstructed to take account of the variability of the possible values of the objects of choice. We shall develop at that point, therefore, what we have termed stochastic utility, or the analysis of utility functions defined over stochastic or random possible outcomes. This again will contribute to the analysis of the financial asset portfolio problem, leading a stage further to the erection of the neoclassical or neo-Walrasian model of financial asset market prices and to the significance that that has for the firm's cost of money capital. The remaining chapters of Part II then take up the cognate questions of the cost of money capital, the controversies surrounding it, and the alternative

20

I Theoretical issues and analytic motivation

recommendations that competing viewpoints provide. A slightly fuller statement will be able to be made at that point regarding the neoclassical criteria for the choice of optimum-investment projects in the firm. In Part III, with the building blocks and some of the superstructure of the neoclassical theory behind us, our attention will turn to alternative constructions. We shall develop in Part III, under the heading of what we have termed "Postclassical perspectives," two main lines of analysis. First, we shall suggest an alternative method of envisioning the interdependence between the production, pricing, investment, and financing decisions of the firm. This will require us to take account of the manner in which the oligopolistic firm, or what Eichner (1976) has expressively called the "megacorp," sets its selling price in such a way as to enable it to generate a cash flow that will contribute to the supply of money capital required for financing investment in the firm. The firm's selling price will be seen to be related to both its production costs and the mark-up necessary to produce the desired flow of internally generated funds. At the same time, interconnections will be seen to exist between the availability of internally generated funds and the supply of money capital, at variously specifiable costs, in the external money capital market. Second, we shall examine more thoroughly in Part III the question of uncertainty, to which we have already referred in a preliminary fashion in this chapter. It will be possible there to set out more completely the manner in which the neoclassical reliance on the probability calculus and probability reduction methods can be replaced by an analysis of "potential surprise" defined over possible, rather than probable, outcomes. From this alternative perspective, a Decision Index for the guidance of investment decisions in the firm will be constructed. It will become clear that the arguments of Part II, "The Neoclassical tradition," and Part III, "Postclassical perspectives," are not completely independent. Many of the building blocks and parts of the conceptual apparatus of Part II will be relevant, though frequently in quite different guises, to the basic thought forms and analytical constructions of Part III. Preparatory to the development of the argument, the following two chapters of Part I will examine the basic questions of (i) the financial statements of the firm and the analytical perspectives that an understanding of them provides and (ii) the meaning and the relevance of the concept of economic value. In the neoclassical tradition, resting as it does on marginal optimization criteria and the maximization of attainable economic values, the problem of valuation is pervasive and stands at the core of the argument. In the last analysis, however, the economic values that can be generated by the firm's activity depend, in the presence of genuine uncertainty, on the rates of return that the suppliers of money capital require. Those suppliers may currently own the shares of stock in the firm and thereby automatically have a claim to the reinvested

1 Thefirmin a monetary economy 21 profits of the firm, or they may be suppliers of money capital funds in the external capital market. The final statements on economic valuation, therefore, will have to be deferred until the full treatment of uncertainty can be taken into account. In the meantime, the arguments presented in Chapter 3 will lay the groundwork and indicate some of the ways in which the relevant concepts have been exploited in the neoclassical tradition.

CHAPTER 2

Assets, capital, and capitalization

The conceptual foundations and technical apparatus of the theory of the firm require an understanding of the financial statements that describe the firm's economic position and structure. These are its periodic balance sheet and income statement. The examination of these documents establishes a linkage between the theory of the firm and an important dichotomy employed in many parts of economic analysis. This is the distinction between stock variables and stock analysis on the one hand and flow variables and flow analysis on the other. On this important matter, the theory of the firm has been ambivalent. Following the recrudescence of interest in the firm's optimization problems in the 1930s, the distinguished economist Kenneth Boulding could still say, in the first edition of his A Reconstruction of Economics in 1950, that "the concept of the balance sheet, unfortunately, has not been employed to any extent in developing the static theory of the firm, so that as generally presented in the textbooks the firm is a strange bloodless creature without a balance sheet, without any visible capital structure, without debts, and engaged apparently in the simultaneous purchase of inputs and sale of outputs at constant rates" (1950, p. 34. See also Boulding, 1966, Vol. I, p. 305). Brian Loasby is probably not too wide of the mark when he comments on "the widespread contempt exhibited by economists for accounting (the more scandalous for not being recognized as a scandal)" (1971, p. 882). If the simple device of the balance sheet, or the firm's economic position statement, had been exploited at an earlier stage in the development of the theory, a clearer understanding would have been gained of the need to distinguish between such assets of the firm as real capital and such liabilities as money capital. But the distinction between real and money capital has not been consistently articulated. A difference of treatment has been accorded the financial and the nonfinancial firm in this respect. The economic significance of the former, for example banks and other financial intermediaries, has stemmed partly from the portfolio of assets in which they have invested. The pools of savings flowing to them from the ultimate savers in the economy have been made available, along with the creation of new money in certain instances, to borrowers in the money capital market. It was natural, therefore, to emphasize the liabilities 22

2 Assets, capital, and capitalization

23

that the financial institutions registered on the receipt of savers' funds and the assets they acquired in their lending and investment activities. The theory of financial intermediation has investigated the criteria of optimum-asset portfolios, having regard to possible trade-offs between prospective rates of return on asset investments and the level of risks associated with them. The theory has examined also the sources from which such institutions acquire funds, in the light, particularly, of the disturbed financial conditions of the late 1970s and early 1980s and the deregulation legislation that occurred at that time (see Vickers, 1985a). The nonfinancial firm, on the other hand, has not generally been accorded such a balance sheet analysis. Here Kenneth Boulding's indictment applies. The economic and the financial theories of the firm have led separate and unsymbiotic existences. The difficulty that static, timeless, equilibrium theorizing has had in finding a place for money led also, paradoxically, to a diminution of the significance of the financial sector in the analytical scheme of things. But there are reasons to believe that progress toward the integration of these different parts of economic analysis has lately accelerated (see Kalecki, 1937; Gabor and Pearce, 1952, 1958; Vickers, 1968; Patinkin, 1965; Davidson, 1978). Stocks and flows in economic theory The stock-flow dichotomy has enjoyed a lively existence in many parts of economic theory. Recall, for example, the intensive debates during the 1930s and 1940s over the correct formulation of the theory of the rate of interest. The differences of view were sharply focused in Keynes's General Theory, in which the rate of interest was understood as the variable that brought into equality the demand for money and the supply of money (Keynes, 1936. See also Robertson, 1940). An alternative formulation saw the rate of interest as the variable that brought into equality the supply of loanable funds and the demand for loanable funds. The first was a stock theory and the latter a flow theory. In due course, it was decided that, on a correct formulation, the two bodies of theory came to the same thing (see Harris, 1981, Chs. 15, 16; Coddington, 1983, p. 76). In its flow aspect, the interest rate was the price of loanable funds. In its stock aspect, it was the price paid for the privilege of holding money. That price was interpreted as the opportunity cost of holding money, or as the income sacrificed by not holding wealth in an alternative income-earning form, such as a short-term government security. The stock-flow distinction was noted earlier also in connection with the interpretation of capital as a factor of production. The production function argument was not the stock of real capital assets held by the firm but Ihzflow of services those real capital assets provided. Further, we can distinguish

24 I Theoretical issues and analytic motivation between the firm's stock of real capital, as that is described and valued in its balance sheet position statement at any given date, and the flow of investment expenditure, which, during any operating period, adds to that existing capital stock. Capital, that is, is a stock concept, whereas investment is a flow concept. During each operating period, a portion of the firm's total capital stock will be worn out by virtue of its contribution to the production activity. This wear and tear, or normal wastage due to use, is referred to in the economic literature as depreciation, or as capital consumption. At the end of each accounting period, a deduction must be made from the firm's gross income stream to cover such depreciation costs. Those deductions are set aside and accumulated during the economic life of the assets so that a fund will be available to replace the assets at the end of their useful lives. Only in this way is it possible to maintain capital intact. The true residual income earned by the firm after making such depreciation deductions is available for distribution to its owners. The stock-flow distinction has application also to the phenomenon of income itself and to the disposability of the firm's income stream. The total money capital invested in the firm by its owners, and the amount of debt or loan capital raised by the firm, stand in its position statement as stock variables, awaiting the payment of interest and dividends and, in the case of the debt, the eventual repayment on the agreed maturity date. A stock of money capital is entrusted to the firm in return for a flow of income payments, adequate in the minds of the suppliers of money capital to compensate them for the risks they bear. Balance sheet, or economic position statement The economic position of the firm at any date is summarized in a balance sheet statement of all of the assets owned and all of the liabilities owed by the firm. Consider the pro forma balance sheet shown in Table 2.1. In the general case, and in an accounting sense, the balance sheet entries will be shown at what are referred to as book values. That is to say, the assets will be recorded at the values at which they were actually acquired by the firm, subject to exceptions we shall note. Similarly, the liabilities will be recorded at the value representing the actual number of dollars received by the firm from the various sources indicated. One such source of money capital is described in Table 2.1 as preferred stock. For our present purposes, this can be taken as a form of equity ownership stock, the differences between preferred and common stock being mainly twofold. First, the preferred stock is usually entitled to receive a fixed annual dividend specified in the preferred stock contract, while the common stock equity holders receive the residue of

2 Assets, capital, and capitalization

25

Table 2.1 Balance sheet Assets Current assets Cash Accounts receivable Inventory Fixed assets

Total assets

Liabilities

XXX XXX XXX XXX

XXX

Current liabilities Accounts payable Short-term debt Long-term capital sources Debt Preferred stock Owners' equity Common stock Earned surplus Total liabilities

XXX XXX

XXX XXX

XXX XXX XXX

income that remains after the payment of preferred stock dividends. Second, the claim of the preferred stockholders will usually rank ahead of the claim of the common stockholders in the event of the insolvency and dissolution of the firm, though it will, of course, rank behind the claim of the debt holders. For those reasons the cost of preferred stock capital will in general be higher than that of the debt capital but lower than the cost of common stock funds. For analytical purposes, and recognizing that preferred stock financing is not used extensively in practice, we shall assume in the following chapters that the sources of money capital can be said to be simply debt on the one hand and common stock, or equity owners' funds, on the other. The debt capital funds may be obtained from either short-term or long-term sources, and in normal economic and financial market conditions the cost of the short-term will be lower than that of the long-term debt. The owner's equity in the firm includes both the common stock account, representing the amount of funds actually received by the firm from the sale of shares of common stock, and an earned surplus account. The latter represents the accumulated value of the earnings retained in the firm during preceding operating periods. The funds corresponding to this entry will have been reinvested by the firm in assets, and they accordingly represent a genuine part of the total ownership investment. In the general case, and provided the outlook for the firm's investment and operating performance is favorable, the market value of the ownership shares of common stock will rise as the total earned surplus retained in the firm increases. The total ownership equity in the firm is sometimes referred to as the firm's net worth. In economic accounting, an entity's net worth is the difference between the total values of its assets and its nonownership liabilities. It would be perfectly possible, of course, to construct a balance sheet or

26 I Theoretical issues and analytic motivation position statement that showed the various accounts at current market values rather than at book or historic values. If commodity prices in general had been rising, the current market value of some of the firm's assets may be greater than their original cost or book values. A profit would then have been made simply by reason that the firm held the assets subject to market value appreciation. If that profit were recorded during a given time period, it would be shown also in a capital surplus account on the liabilities side of the balance sheet, indicating that to that extent the total value of the owners' investment in the firm had increased. Further, the fixed assets referred to in Table 2.1 will be recorded net of the depreciation mentioned earlier. The fixed-asset account will actually be entered at book value or acquisition cost less the total accumulated depreciation that has been written off against the firm's income stream each period since the assets were acquired. If the market value of fixed assets had appreciated as a result of inflationary price changes and if it were desired to record that fact in the firm's economic position statement, the increase in the balance sheet value of the assets would once again have to be offset by a corresponding record of a capital surplus account on the liabilities side. This would again indicate that the value of the owner's investment in the firm had increased. The information conveyed by the balance sheet can be summarized in the following four statements. First, the total of the assets side of the balance sheet indicates the total investment that has been made in the firm. The money capital funds available to the firm have been invested in assets with the objective of generating an income stream and thereby economic values for the owners of the firm. Assets are at work generating income. The economic value of the investment, quite apart from the recorded or book value of the assets, depends on, and is determined by, the level, time shape, and risk characteristics of the income stream it can be expected to generate. Second, the structure of the assets side of the balance sheet indicates the structure of the investment that has been made in the firm. In any given economic environment, there conceivably exists more than one way in which the available money capital can be put to work in an income-earning enterprise. Each of the available investment alternatives can be looked upon as an achievable combination of assets, and the balance sheet shows which of the alternative combinations has been selected by the firm. This immediately raises a twofold question. First, is it possible to establish criteria to determine whether any particular combination of assets can be regarded as an optimum combination; and second, can it be concluded whether the firm's structural and operating decisions, which are reflected in the combination of assets described in the balance sheet, were in fact optimizing decisions? We shall return to this important question. Third, the total of the liabilities side of the balance sheet indicates the total

2 Assets, capital, and capitalization

27

money capital employed in the firm. It follows, of course, that as the total of the liabilities, including ownership capital, precisely equals the total of the assets, we must simply be looking at the same thing from two different points of view. In a sense this is true. But focusing attention on the description of capital employed sharpens the awareness of two significant facts. First, the availability of money capital constitutes a constraint on the investment decisions of the firm and establishes the optimization process in the firm as one of constrained optimization. Second, we are thereby forced to take analytical account of the distinction that this emphasizes between the money capital that is invested and the real capital assets in which the investment is made. Fourth, it follows that in the same way as corresponding questions were raised regarding the assets side of the balance sheet, the following can now be asked with respect to the liabilities side. First, is it possible to establish criteria for determining whether a particular combination of money capital sources is in fact an optimum combination; and second, can it be concluded whether the decisions of the firm that gave rise to the particular combination of liabilities that exists were actually optimizing decisions? Thus, the principal message of the balance sheet is a message regarding the structure of the firm. It describes the economic position of the firm at a point in time. In that sense, it is a static or stationary document. But it is a tool of dynamic analysis when it is recognized that the structure it describes, or the optimum structure to which it might be brought, is itself subject to change as the economic environment changes. Transformation of assets and liabilities: dynamics of the balance sheet Kenneth Boulding early perceived the necessity for an analysis of this kind as an element in the theory of the firm. He has conceived, moreover, of what he expressively referred to as the homeostasis of the balance sheet. By this he means that when, in the course of the firm's operations, it is disturbed from an existing structural state, forces will be set in motion to return it to its previously existing, and conceivably optimum, state (1950, Ch. 2). This notion of homeostasis, however, although it is useful for analytical purposes, "is clearly only a very first approximation . . . [for] it says nothing about what determines the equilibrium state itself" (p. 33), or in our terms, what determines the optimum structure of the firm at any particular time. Boulding's perception, however, enables us to conceive of the operations of the firm as a successive transformation of assets and of liabilities. We conceive, for this purpose, of the firm's internally generated cash flow. First, money capital flows in and is reflected in an addition to the cash account on the assets side of the balance sheet and as a source of investable funds in a debt or equity account on the liabilities side of the balance sheet. As the

28 I Theoretical issues and analytic motivation available cash is spent on acquiring variable factors of production and materials, the cash asset is transformed into an asset called inventories, either of raw materials or work-in-progress or finished goods. At the same time, some of the cash may have been transformed into fixed assets. When in due course the firm's output is sold, the asset of finished goods is transformed into an asset called accounts receivable, and when the purchasers of goods pay their accounts, a further asset transformation occurs to reestablish the cash account. Hopefully, the final receipt of cash will exceed the cash outflow that was necessary to produce the output sold. The excess cash receipts, which will be reflected in the firm's income statement we are about to consider, will also be reflected in an increase in the owners' funds retained in the firm or in dividend payments to them. This is Boulding's homeostasis at work. It can be linked now with the general structure of the firm's income statement. Income statement, or economic performance statement A logical order or priority of ideas exists at this point. Income, we can say, is prior to economic value. The economic value of any asset or investment is precisely defined as the present value, or the present discounted or capitalized value, of the income stream it can be expected to generate during its economic life. We shall return to this important matter of discounting or capitalization. For the present, we observe the general nature of the firm's income statement as shown in Table 2.2. Consider this income statement in the light of the firm's profit function, 7Tt=PtQt-XbitXit-rtDt

(2.1)

where the following notation applies, with all variables being appropriately dated as indicated by the subscript t: IT = net residual income available to the owners p = selling price per unit of output Q = number of units of output sold Xi = quantity of units of factor capacity employed for each factor i incorporated in the productive process bi = input cost per unit of factor capacity r = average rate of interest per annum payable on funds obtained from nonowner sources of capital D = total amount of nonowner funds employed in the firm The equation states that the income available to the owners of the firm is the residual of the total sales revenue minus all factor costs of production and minus also the interest income paid to the providers of debt capital. The sales revenue item in Equation (2.1) is reflected in the first entry in the income

2 Assets, capital, and capitalization

29

Table 2.2 Income statement Total Sales Less Variable factor costs Fixed factor costs Total operating costs Net operating income Less Interest on debt capital Income before tax

XXX XXX XXX XXX XXX XXX XXX

Less Income tax liability

XXX

Net income

XXX

Less Preferred stock and common stock dividends

XXX

Retained earnings

XXX

statement, and the factor costs are reflected in what have been labeled total operating costs. As in the case of the balance sheet, we are again interested principally in the structure of the income statement, meaning thereby (1) the proportionate division of total operating costs between the rewards to factors of production of differing kinds and durabilities and (2) the division of the net operating income between the rewards paid to the providers of debt capital and the residual earnings or rewards available to the owners of the firm. The structure of the earnings stream and of its distribution is related to two principal determining forces. First, the firm's decision regarding the optimum real capital intensity of its production process causes a division between fixed and variable factors of production or between fixed and variable costs of operation. This in turn determines the degree to which the net operating income of the firm will fluctuate when fluctuations in total sales revenue are experienced. The greater the ratio of fixed to variable costs, or the greater the degree of real capital intensity, the greater will be the magnification of fluctuation in net operating income when sales revenues fluctuate. When revenue falls, a greater proportion of it will be absorbed by fixed operating costs, causing a greater than proportionate decline in net operating income. A contrary result will occur when sales revenue increases. Second, the firm's decision regarding the optimum debt-to-equity ratio in its financing structure determines the degree of instability in the residual equity owners' income for given fluctuations in the net operating income. For a relatively large debt-to-equity ratio, a fall in net operating income will cause a greater proportion of it to be absorbed in paying the fixed contractual costs of interest on the debt capital, with, once again, a contrary result when net operating income rises. The concept and the reporting of the firm's net income as summarized in

30

I Theoretical issues and analytic motivation

Table 2.2 may be subject to a number of variables and discretionary management practices that do not call for extended discussion at present. For example, the total operating costs will include, among the variable costs, the value of the materials used in production. The estimate of the value of materials used will generally be equal to the value of the materials stocks at the beginning of the production period plus the value of materials purchased during the period minus the value of inventory stocks at the end of the period. If inventory materials prices rise during the operating period, and if those higher prices are reflected in the valuation of the materials held at the end of the period, this will imply that the materials actually used in production were charged to the income statement at their earlier and lower values. Such a method of materials accounting, referred to as first-in-first-out, or FIFO, will produce a higher reported income in the presence of rising prices, resulting from a lower reported materials cost, than would occur if the opposite, lastin-flrst-out, or LIFO, materials accounting procedure were used. In the latter case, the final inventory stock would be valued at the earlier and lower prices, and the materials used would be included in the cost of goods sold and charged against income at the later and higher prices. Second, a significant element of the fixed operating costs in Table 2.2 is the periodic depreciation charged against the value of the firm's assets. Various methods of depreciation may be employed. Straight-line depreciation writes off and charges against income each year an equal proportion of the asset's original cost. Alternatively, various forms of accelerated depreciation may be used. These have the effect of writing off a larger proportion of the original cost of the asset during the early years of its life. If, once again, asset prices are rising, depreciation by any of the methods suggested will gradually provide a replacement fund equal to the initial cost of the assets, but this may be lower than the actual replacement cost the higher asset prices imply. The accumulated depreciation fund will not then be sufficient to enable the firm to maintain its real capital intact by replacing its assets. In effect, some of its capital value will have been consumed during the time periods preceding the asset replacement because the lower-than-necessary periodic depreciation charges will have led to a higher-than-realistic reported income. The depreciation charge thus affects both the balance sheet, where the asset values are shown at initial cost minus accumulated depreciation, and the income statement, where the reported income is affected in the manner indicated. But the periodic depreciation charge does not, as do other operating costs, give rise to a cash outflow from the firm. Important differences exist between the firm's overall income-generating ability and its ability to produce an internally generated cash flow. For many purposes of analysis, particularly in relation to investment expenditure decisions, it is necessary to focus attention on the cash-flow-generating ability, rather than the prospective income

2 Assets, capital, and capitalization

31

stream, that a project promises. All of these considerations make it necessary for extreme care to be exercised in comparing the reported incomes of different firms and in making decisions regarding portfolio investments and the choices between different firms' capital securities. This requires estimates of the expected income returns and the possible instability or riskiness of those returns that the holding of such securities provides (see Elton and Gruber, 1984; Brealey and Myers, 1984). Operational and financial leverage The first of two basic sources of instability in the firm's income stream is referred to as operational leverage. It results from the interposition of fixed operating costs between the sales revenue and the net operating income. It is reflected in the first half, or the so-called operating section, of the income statement. The second source of instability, which results from the interposition of fixed financial charges between the net operating income and the residual net income, is referred to as financial leverage. It is reflected in the lower half, or the financial section, of the income statement. It will add concreteness to the analysis and will reveal the interconnections that bear on the optimization of both the operating and financial structures of the firm if we consider the information shown in Tables 2.3, 2.4, and 2.5. Here we imagine three firms of the same size, operating in the same industry Table 2.3 Earning ability of firm A Total assets Debt capital Equity capital Financial leverage ratio

$1,000 200 (interest rate 5%) 800 \ Internal rate of return 5%

10%

15%

20%

$50 K)

$100 10

$150 10

$200 10

Income before tax Tax (50%)

40 20

90 45

140 70

190 95

Net income

20

45

70

95

2.5

5.6

8.7

11.9

Net operating income Interest on debt

Rate of return on equity, %

32

I Theoretical issues and analytic motivation

Table 2.4 Earning ability of firm B Total assets Debt capital Equity capital Financial leverage ratio

$1,000 500 (interest rate 5%) 500 1 Internal rate of return

Net operating income Interest on debt

5%

10%

15%

20%

$50 25

$100 25

$150 25

$200 25

Income before tax Tax (50%) Net income

25 12.5 12.5

75 37.5 37.5

125 62.5 62.5

175 87.5 87.5

Rate of return on equity, %

2.5

7.5

12.5

17.5

Table 2.5 Earning ability of firm C Total assets Debt capital Equity capital Financial leverage ratio

$1,000 800 (interest rate 6%) 200 4 Internal rate of return

Net operating income Interest on debt Income before tax Tax (50%) Net income Rate of return on equity, %

5%

10%

15%

20%

$50 48 2

$100 48 52 26 26

$150 48 102

$200 48 152 76 76

13

25.5

1 I 0.5

51

51

38

and the same general economic environment, having similar operating structures, and earning the same rate of return on their total investment. The latter, or the internal rate of return, may be expressed as the percentage of net operating income to the total capital employed, or total assets at work in the

2 Assets, capital, and capitalization

33

Table 2.6 Summary of financial leverage relationships Degree of leverage

Low Medium High

Rate of return on equity for indicated internal rates of return 5%

10%

15%

20%

2.5 2.5 0.5

5.6 7.5 13.0

8.7 12.5 25.5

11.9 17.5 38.0

Percentage variation in rate of return on equity for indicated changes in internal rates of return

Low Medium High

Internal rate of return changing from 10 to 5% 10 to 15% = -50% = +50%

10 to 20% = +100%

-55 -66 -96

112 133 192

55 67 96

firm. Imagine, however, that the structural difference between the firms is that they have adopted the differing financial structures indicated in the tables. It is to be expected that a firm will have to pay a higher rate of interest on its debt capital as its financial leverage or debt-to-equity ratio increases, due, of course, to the increased riskiness of the debt holders' position. A nod in the direction of realism is therefore made in Table 2.5 by supposing that the rate of interest has risen from 5 to 6 percent in response to the sharply higher financial leverage ratio. In order to clarify the financial leverage effects in the briefest space, relevant data from Tables 2.3, 2.4, and 2.5 are summarized in Table 2.6. In this table, firms A, B, and C are referred to as low-, medium-, and high-leverage firms, respectively. Several summary propositions may be adduced from the data in Table 2.6. First, in both the low-leverage and medium-leverage firms the rate of return on equity is 2.5 percent when the internal rate of return on total capital employed is 5 percent. This follows from the fact that the internal rate of return is precisely the same as the rate of interest being paid on the debt capital. No surplus earnings are therefore available to accrue to the residual owners. Only if the debt capital, when it is put to work in the firm, should earn a rate of return greater than the rate of interest being paid on the debt can the equity owners be said to receive the benefits of financial leverage. But even in the case where no change in the owners' income occurs from a leverage effect, their overall economic position could in fact be worsened. This could occur if the potential instability introduced into their income ex-

34

I Theoretical issues and analytic motivation

pectations by the debt-to-equity ratio should induce an increase in the equity market's capitalization rate, or required rate of return. For this would cause a reduction in the capitalized value of their investment in ways that we shall analyze more fully. The example before us shows clearly how such effects may emerge. If the internal rate of return should be 5 percent while the rate of interest on borrowed funds should be, say, 6 percent, as in the case of the high-leverage firm, the contractual interest on the debt would absorb part of what would otherwise be the residual income of the owners, and the rate of return on the equity could fall to a very low level. Second, in those instances in which the internal rate of return is greater than the interest rate on the debt, the rate of return on equity will be greater where the degree of financial leverage is greater, or where the debt-to-equity ratio is higher. This is shown in the last three columns of the first part of Table 2.6. Third, for any given proportionate change in the internal rate of return, the existence of the leverage effect will give rise to a magnified change in the rate of return on the equity capital. Consider the last two columns of the lower section of Table 2.6. Increases of 50 and 100 percent in the internal rates of return caused larger rates of increase in the rate of return on equity in each of the low-, medium-, and high-leverage firms. Even more important, the degree of magnification of residual rates of return is seen to increase as the financial leverage ratio increases. But, of course, the same magnification effect can work in reverse if the underlying income-generating ability of the firm should fall. The reverse effect is depicted in the first column of the lower section of Table 2.6. Economic decisions in the firm: production, capital, and finance The theory of the firm has conventionally regarded the objective of the firm's management as that of maximizing the attainable profit under both short-run and long-run conditions. In the short-run analysis, the real capital stock of the firm has generally been taken as fixed. The operating decision problem was that of settling on the optimum amount of variable factors to employ in conjunction with the given plant and other elements of fixed capital. In the long run, on the other hand, all factors of production, including those that were taken as fixed in the short run, were assumed to be variable. In the long run, capital was assumed to be mobile. It could, it was imagined, migrate to other and more attractive lines of economic employment. To the extent that the theory broke out of the static mold that this summary of the short- and longrun comparison envisages, a legitimate target of management might be that of maximizing, or optimizing subject to the risks involved, the rate of growth of the firm. This is taken as the primary objective of the firm in many expres-

2 Assets, capital, and capitalization 35 sions of what we have termed the postclassical theory. In other cases, the objective might be that of maximizing the market share of the firm, subject, in some instances, to the realization of an acceptable rate of return on the capital employed. Inadequate attention has usually been given, however, to the question of precisely what was thought to be mobile in the long run but assumed to be given and fixed in the short run. If it was real capital that was mobile, insufficient attention was given to the fact that the specificity of real capital assets diminished their economic and functional mobility. There is little reason to believe that a ready market exists for second-hand real capital assets that were initially designed for a specific economic use. If, on the other hand, it was assumed that money capital was mobile, it then became necessary to explain precisely how the money value of immobile real assets could be realized in such a way as to permit that money capital to migrate to alternative uses or other investment opportunities. The truth of the matter is that the money value of capital assets depends on the prospective income streams they are capable of generating in their existing or alternative uses. If the income-generating ability of an asset should, for any of a number of economic reasons, evaporate, the economic value of that asset will be reduced to zero, and its monetary value, if any, will be determined simply by its disposal or scrap value. The linkage that the theory of the firm has imagined to exist between the short and the long run might not, therefore, exist. It is necessary to recognize that while money capital is homogeneous, real capital is a sum of heterogeneous units. Lachmann, in an early formulation of related theoretical problems, observed that to invest in a firm is to "de-homogenize money capital" (1956, p. 36), and he warned against the danger to analysis of what he called an "illegitimate generalization based on the homogeneity hypothesis" (p. 10). The theory of the firm needs to distinguish between two different kinds or levels of analysis so far as the capital investment and the capital migration problems are concerned. First, we can envisage an ab initio planning problem in connection with which we can construct models of optimum enterprise structure. In that case, the theory of the firm determines the optimum relations between fixed and variable factors of production with which to produce an optimum level of output or an optimum product mix. Second, we can envisage a problem of sequential structural planning for a firm that is an established going concern. In that case, we need to understand how, at each sequential planning date in real historic time, the money capital availability that constrains the firm's structural decisions takes account of the money capital inherent in the true economic value of its assets. In some instances, claims to all or a portion of the assets may be able to be sold in the capital securities market and the funds made available for reinvestment. Or where appropriate,

36 I Theoretical issues and analytic motivation account may be taken of the money capital available from the liquidation value of those assets that can no longer be optimally employed in the firm. The theory of the firm has generally failed, however, to distinguish between these two very different capital investment problems, and analytical refuge has been taken in the assumption of perfect capital mobility. More completely, realism requires us to distinguish between (i) the ab initio planning decision, (ii) the short-run operating decision, and (iii) the sequential restructuring decision of the firm. The last mentioned might lead simply to an expansion of existing facilities for the existing or closely related purposes or to a partial or total liquidation and reinvestment. In Chapter 4 we shall establish a method of looking at the first of these decision problems. In doing so, we shall maintain the distinctions between homogeneous money capital and the heterogeneous units of real capital to which money capital is committed, at the same time as we envisage the linkage and interrelations between them. For this purpose, it is useful to conceive of a trilogy of decision problems confronting the firm. We refer to them as (i) the production decision, (ii) the investment decision, and (iii) the financing decision. The interdependence between the elements of this trilogy, the solution values of which will follow from the maximization of the value of an appropriately defined objective function, can be seen in the following way. First, the production decision is concerned to discover (i) the optimum level of output, or the optimum level and combination of outputs in a multiproduct firm, and (ii) the optimum combination of factor inputs, both capital and variable factors, that should be employed, given the state of technological knowledge and the implied technological possibilities in the production function, to produce that optimum output. The product output decision will involve also, of course, a decision as to the market price at which the output should be sold, given the general assumption of imperfect competition or an average revenue function of the form employed in the following chapter, p^piQ). When, at a later stage of our work, fuller account is taken of uncertainty, the optimum size and structure of the firm may well be such as to afford the firm a flexible range of short-run operations. It may be desirable, in other words, to build into the firm a target level of standard excess capacity. The decision to produce implies the presence in the firm of a production apparatus. The investment problem, therefore, is concerned to discover the optimum amount and combination of real and monetary assets in which to invest in order to establish and maintain the productive process that is necessary to produce the optimum output with the optimum combination of factor inputs. But the decision to invest in this manner implies the need to finance the investment. It follows that the financing problem is concerned to discover the optimum combination of financing sources, or sources of money capital, that

2 Assets, capital, and capitalization 37 should be used to finance the optimum investment in assets necessary to maintain, at their interdependent optimum values, the size, structure, and operating processes of the firm. This financing decision may be subject, however, to whatever degree of unused borrowing capacity, or flexibility for varying possible future loan operations, it might be desired to maintain. The interdependence of the elements of the firm's structural decision trilogy will become clear. Caution must be raised against too easy an assumption of the efficiency of sequential rather than mutually determinate decision making in these important respects. Economic decisions in the firm and economic position and performance statements In the light of our discussion of the firm's economic balance sheet and its income or economic performance statement, we may visualize the manner in which the solutions to this trilogy of decision problems are reflected in those analytical documents. The outcomes of each of the decision problems will be reflected in two different ways in the economic statements, as reference to Tables 2.1 and 2.2 in connection with the following points will confirm. The perception of the relations between them will clarify further the interdependence between the decision problems themselves. First, the production problem envisages a level of production and sale of output, and the sales revenue, or the level of output times the unit price at which it is sold, will appear at the head of the income statement. The production problem decision envisages also an optimum combination of factor inputs. The costs incurred by these will be reflected in the income statement as either variable or fixed operating costs. In the case of the fixed factors, it will be necessary to determine the periodic operating flow costs of real capital, including, as we have observed in a preliminary fashion already, the periodic depreciation costs. We have already noted that the structure of the operating section of the income statement will reflect the degree of operational leverage at work in the firm. The solution to the production problem will be reflected also in the balance sheet, and here it illustrates an aspect of Boulding's homeostasis. To the extent that the production decision involves an optimum combination of variable and fixed factors of production, it will be necessary to invest in real capital assets to provide the periodic flows of fixed factor services. It is the latter, we recall, that we envisage as arguments in the firm's technological production function. That investment in real capital assets, then, will be reflected in fixedasset accounts in the balance sheet or economic position statement. Second, the solution to the investment problem will be reflected in both the balance sheet and the income statement. We have just seen that the decision

38

I Theoretical issues and analytic motivation

to invest in fixed assets will be reflected on the assets side of the balance sheet. But we recall also that the investment problem is concerned with the optimum combination of real and monetary assets, and its outcome will be reflected also, therefore, in the current-assets section of the balance sheet, in the cash, accounts receivable, and inventory accounts. Additionally, the investment problem decision will be reflected in the operating section of the income statement for a reason that may escape attention because it does not give rise to immediate cash flows in the firm. We refer again to the manner in which the decision to invest in depreciable assets implies a periodic depreciation charge as an inclusion in costs against the firm's income stream. Third, the solution to the financing problem will again be reflected in both the balance sheet and the income statement. To the extent that the financing decision determines an optimum combination of sources of money capital, envisaging a mix of debt and equity capital, the outcome will be reflected on the liabilities side of the balance sheet. Further, the need to make a fixed periodic interest payment on the debt capital will be reflected in the financial section of the income statement. The operating section of the income statement, it will be recalled, describes the income-generating ability of the total assets at work in the firm, and the financial section describes the manner in which the net operating income is distributed among the providers of the money capital funds. A portion, which constitutes a prior claim against the income stream, is paid to the debt holders, and the remainder, after the payment of taxes, becomes the residual earnings of the equity owners. Economic valuation, or capitalization of income streams Equation (2.1), in the context of the firm's income or economic performance statement, describes the periodic net profit that becomes the property of the residual equity owners. The theory of the firm has generally focused on this net profit as the maximand variable in the firm's optimization problem. It has, moreover, as Equation (2.1) implies, usually treated the profit problem in a static sense. It has assumed that when the firm is brought to an equilibrium posture and earnings position, the net profit will recur during each future operating period. For many reasons, however, the profit criterion is inadequate. But its deficiencies can be emphasized only when the questions of money capital and the lapse of real economic time are integrated into the firm's decision problems. The firm must be interested not simply in the net profit that may be generated and become the property of the owners but also in what we shall call the economic value of that income. For it is this that determines the economic value of the owners' investment in the firm. In short, we replace the traditional objective of profit maximization by that of economic value maximiza-

2 Assets, capital, and capitalization 39 tion. That means, at one remove, the maximization of the market value of the firm's outstanding shares of common stock. To achieve this objective, it may be necessary to refrain from planning for the maximum attainable economic profit. For it is necessary to take into account also the risks being borne in the pursuit of that profit. The suppliers of money capital, given the assumption of risk aversion on the supply side of the money capital market, will in general require a higher rate of return to compensate them for undertaking higher degrees of risk. The objective of the firm, then, is to generate that income stream for the owners that, when it is valued at a capitalization rate equal to the owner's required rate of return, will attain a maximum possible value. Clearly, some difficult problems associated with the specification of that appropriate capitalization rate will need to be addressed. It will give concreteness to the transition from profit maximization to economic value maximization if the following summary relationship is observed. Consider the profit variable defined in Equation (2.1), 777, describing the firm's periodic net income accruing to the owners. The economic value of such an income stream over time, or, that is, the economic value of the ownership, V, will be the present discounted value, or the present capitalized value, of the income stream when that income stream is discounted at a capitalization rate equal to the owners' required rate of return. We may assume at this stage that this capitalization rate, p, is determined by money capital market conditions and will in general be higher or lower depending on whether the degree of risk to which the equity owners are exposed is higher or lower. We write p = p (risk) to indicate the dependence of the required rate of return on the risk being borne, and we shall in due course observe the possible functional forms of that dependence. We can therefore write (22)

or V = f*7r,e-'>ldt

(2.3)

Equation (2.2) involves the familiar assumption of discontinuous discounting, whereas Equation (2.3) implies continuous discounting. In the simple case where the profit is assumed to remain constant through all future operating periods, and noting that the residual equity shares whose value we are here specifying have no maturity date and that they therefore earn in effect a perpetual or infinitely long income stream, Equations (2.2) and (2.3) both reduce to the straightforward expression V=7r/p

(2.4)

40

I Theoretical issues and analytic motivation

In these expressions, the symbol V refers to economic value. In the more complex cases that will engage us, the economic value will need to be interpreted as containing within it a greater or lesser degree of uncertainty. In the conventional theory in which uncertainty is interpreted as probabilistically reducible risk, the economic value will be interpreted as a random variable by virtue of its dependence on the time vector of periodic profit, the elements of which are also understood as random variables. The received traditions in this regard require us to bring into service the notions of the probability calculus. In the following chapter we shall examine the fundamentals of economic valuation more fully. Then in the chapters that follow, in Part II, we shall expand the neoclassical theory to specify a firm's objective function that takes full account of the money capital availability constraint. This will lead to a more complete demonstration of the mutually determinate production, pricing, investment, and financing decision solutions.

CHAPTER 3

The concept and relevance of economic value The concept of income is logically prior to the concept of value. The economic value of an asset depends on the future stream of benefits that the asset will, or is expected to, provide. In the case of an asset, such as we are preponderantly concerned with in the theory of the firm, which is expected to earn a specifiable amount of income or realize a definable cash flow in the future, its economic value is defined as the present value, or the present discounted or capitalized value, of that prospective income stream. Our purpose in this chapter is to make these concepts related to economic value more precise and thereby lay the foundation for visualizing the determinants of value in more complex cases. In this chapter we shall not give an adequate account of the relevance to the valuation problem of the presence of risk or uncertainty. That important issue must await the subsequent development, in the neoclassical case, of the apparatus of probability and its application to the definition of risk and, in the later postclassical case, the development of the potential surprise analysis. Economic valuation rests on the proposition that a dollar available today is worth more than a dollar available tomorrow or next year or at some other time in the future. This is not simply a matter of saying that a bird in the hand is worth two in the bush. For we may suppose that the expectation of receiving a dollar in the future may be held with absolute certainty. There may be no risk or uncertainty involved. In the ordinary course of affairs, a dollar received today would still be worth more than a dollar to be received in the future, for it could be invested for the period between the present and the date on which the future expected dollar will become available. The investment will then earn income in the form of a rate of interest, and when the date of expectation arrives, its accumulated value will be greater than one dollar. We can therefore envisage what we shall call differently dated money values. We shall speak of present-dated money and future-dated money. Furthermore, we may speak of the future value of present-dated money and the present value of future-dated money. These concepts may be brought together and their relationships exhibited under the heading of what is referred to as the time value of money. 41

42

I Theoretical issues and analytic motivation

Time value of money Consider an amount of present-dated money, referred to by the symbol P, which is invested for one year at a rate of interest of r percent per annum. The value of the investment at the end of the year, here designated as W\, can be stated as Wx=P + rP = P(\+r)

(3.1)

This example assumes that the interest on the investment is added only once, at the end of the year. It thereby reflects the most extreme form of what is called discontinuous interest imputation. If the investment were left to accumulate for another year, the value at the end of the second year could be similarly described. It would be equal to the value that the investment had attained by the end of the first year plus interest at the assumed rate of r percent per annum on that amount: W2 = Wi(l+r) = P ( l + r ) ( l + r ) = J P(l+r) 2

(3.2)

It follows that if the investment is left to accumulate for any desired number of years, say t years, the value of the investment at the end of that time could be stated as1 Wt = P(\+rY

(3.3)

In economics, and particularly in connection with the problem of economic valuation, we are frequently interested not in the future value of a presentdated sum but in the present value of a future-dated amount of money. Present values of future-dated sums can be derived directly from the general expression contained in Equation (3.3). It follows by transposition that P

Or

P

= W,(l+r)-'

(3.4)

This procedure of reducing a future-dated sum to its present-value equivalent is referred to as "discounting." A more formal definition of present value is the following. The present value of a future expected sum is that amount of money that, if it were invested now at a specified compound rate of interest per annum, would amount to that future sum on the future designated date. 1

The corresponding form of Equation (3.3) if continuous rather than discontinuous interest imputation and growth is assumed is Wt = Pe", where the symbol e is a mathematical constant whose value approximates 2.71828. In this chapter we shall present the analysis in its discontinuous form. If the continuous forms were used, the discontinuous summations in the following section could be written as, for example, LS#~rt in place of 25,(1 + r)~', or the integral fS(t)e~" dt could be employed. Forms of this kind will appear in Chapter 4, where more extensive use is made of the differential and integral calculus.

3 Concept and relevance of economic value

43

Present values of future expected series of money payments Economic valuation is frequently concerned not simply with the present value of a future-dated sum but with the present value of a series of such sums. The problem can be visualized in the valuation of an income-earning asset such as a government bond. Imagine such a security with a face value of $1,000, the amount, that is to say, written on the face of the security and the amount that will be repaid to the holder when the bond matures and becomes due for repayment. Suppose that stated on the face of the bond also are words to the effect that the holder is entitled to receive an interest payment each year equal to 6 percent of the face amount of the bond. That stated rate of interest is referred to as the coupon rate. It implies that the holder will receive $60 per annum (6 percent of the $1,000 face amount) so long as he holds the bond. We can suppose, to complete the example, that the bond will mature and will be repaid (or redeemed) 10 years from the present date. The question now arises as to what value the bond would have in the capital asset market if the current rate of interest were, say, 5 percent. In that case, investors in general will be prepared to purchase and hold the bond only if they can acquire it at a price that will make it possible for them to realize a rate of return of 5 percent on their investment. Five percent is, in such a situation, the investor's opportunity cost of the investment. By purchasing the bond, the investor would be assured of receiving in exchange for his outlay the total of (i) $60 interest payment each year for the next 10 years plus (ii) the amount of $1,000 as the redemption value of the bond when it matures 10 years from the present. We calculate the present value of these amounts, and therefore the economic value of the bond, by applying the familiar present-value rules. Using the notation Vo to refer to the present value, we have V0 = $60(l + 0.05)- ! +60(1+ 0.05)- 2 + 60(l+0.05)- 3 + ---+60(l+0.05)- 1 0 +1,000(1+0.05)- 1 0

(3.5)

In general terms, such an expression can be written as follows, where St describes the annual interest cash inflow, M the maturity or redemption value, r the discount rate or the required rate of return at which the expected cash flow stream is being discounted or "valued," and n the number of years between the present and the maturity date:

Vo= £ s , ( l + r ) - ' + M(l+r)-"

(3.6)

t=\

The present value of the series of expected future receipts is simply the sum of the present discounted values of each of the separate elements in the expected cash flow when those cash flow elements are discounted at a discount factor equal to the investor's required rate of return.

44

I Theoretical issues and analytic motivation

Application to the firm's investment decisions The logic of the firm's investment decision can be exhibited by employing the following notation and making an application of the valuation principles that have just been adduced. St — expected cash inflow attributable to the proposed investment project in time period (or year) t n = number of years in the expected economic life of the project m = the firm's cost of money capital k = expected true rate of profit on the project V= economic value, or present capitalized value, of the project C = required total money capital outlay on the project Jn = liquidation value of the asset investment at the end of the economic life of the project Recalling the definitions introduced in connection with the firm's balance sheet or economic position statement and its income or economic performance statement in Chapter 2, only a minimal further comment on the foregoing symbols is necessary. First, the periodic cash inflow, St, is to be regarded as the enterprise net incremental cash flow properly attributable to the project being examined. It is derived by considering the total cash inflow that, it is estimated, would be generated by the firm as a whole if the project under consideration were adopted and then deducting from this the cash inflow that would be generated if the project were not undertaken. The difference will be the marginal cash flow due to the presence of the new project in the firm. The cash flow should be taken net of all operating costs but before the deduction of depreciation on the real capital assets. The reason for the latter is that in order to be economically worthwhile the cash inflow should be large enough to enable the firm to realize two objectives: (i) to set aside out of that cash flow each year an amount, by way of a contribution to a sinking fund, such that at the end of the life of the real assets employed in the project that sinking fund will be sufficiently large to provide for the replacement of the assets; and (ii) to provide, after the setting aside of that depreciation allowance, an acceptable rate of return on the money capital employed. Only if both of these objectives are realized will the capital committed to the investment be preserved intact, at the same time as the desired rate of return is realized. Second, the firm's cost of money capital, here designated m, describes the estimated cost of capital that the firm will be required to pay in order to acquire the money capital funds necessary to finance the project. This money capital may be obtained by retaining in the firm, rather than distributing as dividends, a portion of the previously generated income. Or it may be ob-

3 Concept and relevance of economic value

45

tained by making new capital issues, by either borrowing or raising new equity, in the external money capital market. In some sense, yet to be explored more fully, the firm's marginal cost of capital, m, will be an estimate of the risk-adjusted rate of return that the suppliers of money capital require in order to induce them to supply money capital to the firm and to accept the degree of risk that that involves. We employ this discount factor in the present example even though, as we indicated at the beginning of this chapter, we are not endeavoring to make full adjustments for risk and uncertainty at this stage. Our concern here is simply with the logic and mechanics of economic valuation, and both the future expected cash flows and the rigorously relevant discount rate will be more fully specified in the chapters that follow. Third, the required money capital outlay on the project, here designated C and assumed to be made in one lump sum at the initial implementation of the project, should take account of all necessary expenditures that have to be made in order to bring the project to operating or cash-flow-generating condition. It is comprised of three elements: (i) the cost of acquiring the real capital assets necessary to implement the project, (ii) the installation expenses incurred prior to the operational date and properly attributable to the project, and (iii) the value of the firm's incremental investment in current assets, such as inventories and accounts receivable, which has to be made in order to maintain the project in continuous operating and income-generating condition. The income stream generated by the project must be sufficiently large to provide an acceptable rate of return on all capital expenditures incurred under these three heads. The economic value of the investment project, or what we have referred to as the present discounted value of the future expected cash inflow stream, is then estimated by discounting those cash flows at a discount factor equal to the firm's cost of money capital. The expected liquidation value of the project at its terminal date, which is referred to as Jn in Equation (3.7), will in general include at least the nondepreciable portion of the current-asset investments included in the total capital outlay on the project. It follows that

V= i « l + m ) - ? + yn(l+m)-"

(3.7)

t=\

This provides an investment decision criterion in the form of a comparison between the economic value of the project and the money capital outlay that is necessary to bring the project to operational condition: VgC

(3.8)

It follows that the higher the discount factor in Equation (3.7), or the higher the cost of money capital, the lower will be the economic value of the project.

46

I Theoretical issues and analytic motivation s

NPV

NPV

NPVft

Figure 3.1 The net present value of the project, or the difference between the economic value and the money capital outlay, NPV-V-C

(3.9)

similarly provides the investment decision criterion NPV SO

(3.10)

A net present-value function showing the dependence of the NPV on the discount factor (cost of money capital) may be described as in Figure 3.1. This figure shows the net present-value functions for two separate investment projects, A and B. Each is negatively inclined as inferred from the preceding analysis. It could be imagined that projects A and B required the same money capital outlay and had the same economic life but that the time shape of the cash inflows was different for project A from that for project B. For higher discount factors, NPVA falls more rapidly than does NPV#. This would occur if project A cash inflows were expected to be higher in the later years of the life of the project. In that case, the higher discount factor would cause a larger reduction in the NPV by reason that it was applied to larger nominal magnitudes for a longer period of time. At a cost of capital equal to 0A in Figure 3.1, NPVA is equal to zero, and a comparable condition occurs at a cost of capital OB in the case of NPVfi. To interpret these data, we can inspect an alternative formulation of the firm's decision problem.

3 Concept and relevance of economic value

47

Employing familiar discounting procedures, it is possible to estimate the discount factor that must be employed in order to make the present discounted value of the future expected cash inflows equal to the required capital outlay on the project. This discount factor is shown as k in C= £ St(\+k)-' + Jn(l+k)-n

(3.11)

The solution value of k is referred to as the project's true rate of profit or the internal rate of return. If, for simplicity, we assume that the terminal value is zero and that the periodic cash flow is constant, the solution value of k may be written as (3.12) For projects of longer economic lives, or as n in Equation (3.12) increases, the true rate of profit approaches asymptotically the value SIC. This will be recognized as the reciprocal of the project's payoff period. The latter, or CIS, is the number of years it would take for the project's cumulative annual cash inflows to aggregate to the initial capital investment outlay. Given the computation of the true rate of profit in the investment project, the investment decision criterion can be established by comparing the rate of profit with the firm's marginal cost of money capital: k%m

(3.13)

For an investment project in which a capital outlay in the present is followed by a positive return cash inflow each year during the life of the project, the criteria derived in Equations (3.8), (3.10), and (3.13) will all provide the same answer to an accept-or-reject decision. Returning now to Figure 3.1 and recalling the definition of the true rate of profit as the discount factor that equates the economic value of the cash inflows to the money capital outlay, or the NPV to zero, the abscissa intercepts in Figure 3.1 can be understood as describing the true rates of profit in the respective investment projects. The possibilities shown in Figure 3.1, however, illustrate the difficulties in using the investment decision criteria as ranking devices when it is desired to specify the order of ranking or desirability of different projects. If, for example, the rate-of-profit criterion were used, project B would be ranked ahead of, or would be preferred to, project A. But if the NPV criterion were used for ranking purposes, no single unambiguous answer would be provided. For at discount factors higher than m*, project B has a higher NPV than does project A. But at discount factors below m*, the opposite ranking occurs. We have, then, the potential problem of inverse rankings at low costs of capital.

48

I Theoretical issues and analytic motivation

In such an event, the economically correct order of preference will always be given by the economic value criterion, for this, as will be seen immediately, involves a more conservative reinvestment rate assumption. The reinvestment rate assumption is understood by referring again to the solution value of the true rate of profit k in Equation (3.11). This, we recall, is derived from the periodic cash flows of St. The cash flow elements, in turn, can be regarded as divisible into (i) a "return-of-capital" component that is, as we have seen, placed into a sinking fund to provide for the replacement of the real assets involved in the project and (ii) a "true residual income" component. In order to conclude that the solution value of k is in fact a true rate of profit, the return-of-capital components must be able to be reinvested at the same cumulative annual rate of return as k, the rate being earned on the original project. Consider for the moment that k is in this rigorous sense a true rate of profit. Then the residual income generated by the investment must be regarded as k percent of the original capital committed to it. Taking kC, therefore, as a measure of the true income component of the cash flow, the balance, or S — kC, must be regarded as the return-of-capital component available to preserve the original capital investment intact. If such an amount, S — kC, is reinvested at the end of the first year of the life of the project and accumulated at k percent per annum, its accumulated value at the terminal date of the project will be equal to (S — kC) (1 + k)n ~l. If all such annual reinvestment components were similarly accumulated, the condition it is desired to attain by the terminal date would be described as C= £ (S-kC)(l +k)n-f t=

(3.14)

I

It can be shown by algebraic manipulation that the same value of k satisfies both Equations (3.11) and (3.14). If, therefore, there is reason to believe that the periodic return-of-capital components of the cash flow cannot be reinvested at as favorable a rate as appears to be earned on the original project, then in order to maintain the capital investment intact, the annual return-ofcapital components will have to be larger. In that event, the residual income components will be smaller, and the true rate of profit will accordingly be lower than was initially anticipated. The amount by which the true estimate of the rate of return must then be reduced will depend on the assumption that is made regarding the attainable reinvestment rate. A project's NPV function will be monotonically negative as in Figure 3.1 only in connection with what we refer to as a normal investment project. This may be defined as a project for which a cash outlay in the present is followed by a positive cash inflow each period in the future. Correspondingly, a nonnormal investment project will be one in which the cash inflows are negative

3 Concept and relevance of economic value

49

s

NPV

Figure 3.2 in one or more years in the future. A negative cash inflow may be caused by the need to make unusually high maintenance expenditures in a particular year. When a change of sign in the elements of the cash flow vector occurs, it can be shown that the NPV function will cross the abscissa, as in Figure 3.2, for example, as many times as the number of changes of sign in the cash flow elements. This is because the solution value of the true rate of profit, or k in Equation (3.11), is provided by the positive roots of an wth-degree polynomial for a project whose cash flow extends for n years. For a "normal" project as defined above, the solution values of these roots converge on one positive real magnitude. Multiple solution values, or multiple roots of the relevant polynomial, occur in the "nonnormal" case. Consider a project that had a life of two years with a cash outflow to finance the project occurring at the beginning of the first year followed by a cash inflow at the end of the first and second years. This would mean that the true rate of profit would be provided by the positive root of a second-degree equation. The cash flow would contain one change of sign, negative at the beginning of the first year (the investment outflow) and positive at the end of the first and second years (the cash inflows). This one change of sign would provide one positive real root, described in Figure 3.1 by the abscissa intercept. An investment outlay today of $100, for example, may be followed by cash inflows of $10 and $110, respectively, one and two years from the present date. Equation (3.11) would then specify the rate of profit as the positive solution value of k in the equation

50

I Theoretical issues and analytic motivation

100= 10(1 +k)~l + 110(1 + k)~2. This requires a solution of the equation 100( 1 + k)2 = 10( 1 + k) + 110 or the second-degree polynomial 100k2 +\90k20 = 0. This provides a positive root of 10 percent. That is the rate of profit in the project. In the example exhibited in Figure 3.2, the project will display positive net present values for costs of capital between mx and m2. But at costs of capital or discount factors outside of that range the NPV is negative. We then have what is referred to as the problem of multiple roots, or multiple solution values for k in the rate of profit equation. Given that condition, the appropriate investment decision can be made by discounting the cash flows at a specified cost of capital and formulating the problem in terms of the economic value criterion. That criterion makes the conservative assumption that the cash flows can be reinvested at a rate of return equal to the cost of capital, a less stringent assumption than is made in that respect by the rate-of-profit criterion. If, as we have envisaged, the cost of capital is interpretable as an opportunity cost, it is reasonable to assume that reinvestment opportunities will be available at that rate (see Lorie and Savage, 1955; Solomon, 1956). Economic value and the relevance of separation theorems The logic of the preceding analysis points to a significant proposition. Provided it is possible to specify an opportunity cost of capital or a required rate of return, or a relevant discount factor or capitalization rate at which futuredated sums may be discounted to their present values, optimization decisions should be made by maximizing present or economic values. This has become a widely accepted rule in the economics of financial optimization. It has been argued that on the assumption that borrowing and lending of money capital can take place at a designated rate of interest, intertemporal production and consumption decisions can be "separated." The so-called separation theorems that have entered the literature are highly relevant to the neoclassical (or neo-Walrasian) equilibrium theory of financial asset prices and to the significance those prices have for the firm's cost of money capital. In preparation for our encounter with that level of argument in Part II, we may look at two reasonably straightforward applications to this question of the economic valuation apparatus we have examined in this chapter. In this exposition, we shall make use of the neoclassical assumption that unlimited borrowing and lending opportunities exist in perfect capital markets and that no single borrower or lender is able, by his individual actions, to exert any impact on the assumedly equilibrium rate of interest that exists. Consider for this purpose an individual who (1) possesses a given resource endowment at the present time, t0, and who has the sure prospect of a future income endowment accruing at the end of one time period from the present, at t\; (2) is confronted with definable intertemporal production opportunities

3 Concept and relevance of economic value

51

Figure 3.3 extending over the same single time period; and (3) has the opportunity of borrowing or lending over the same time period at a given and exogenously determined market rate of interest. Such a situation is described in Figure 3.3. In Figure 3.3, the axes labeled t0 and t\ represent resource availabilities at each of the two time dates. The point E indicates that the individual's time vector of endowments is described by RQ and R\. Through the endowment point, an intertemporal production possibility or production transformation frontier, PPf, is described. This indicates that by consuming less than his given endowment in the current period and devoting part of his resources to production, a larger amount of consumable resources could be made available to the individual in the following period. The extent to which this would be economically advisable would depend on his intertemporal consumption preferences, on the intertemporal production possibilities, and as we shall see, on the externally given market rate of interest. Let us assume that a market rate of interest r is implicitly defined by the slope of the valuation line intercepting the to axis at Wo. The magnitude of Wo then describes the value of the individual's two-period endowment vector, measured in present-value magnitudes, by virtue of the fact that the valuation line passes through the endowment point E. In other words, Wo = Ro + R\/(l+r)

(3.15)

and every point on the same valuation line would have the same present value.

52

I Theoretical issues and analytic motivation

An indifference curve through the endowment point E, labeled /, indicates that E does not lie on the highest attainable indifference curve. For by moving along the production transformation frontier from E in the direction indicated by the arrow, the individual can climb on to higher indifference curves. He would then be investing part of his currently available resources in production activities, the output of which would accrue in the following period. The optimum extent of such an intertemporal resource reallocation will depend on the given rate of interest. The production possibility frontier is crossing a family of valuation lines that are parallel by virtue of the given rate of interest, and the optimum production point will be P* in the diagram. At that point, the production possibility frontier is tangential to the highest attainable valuation line. Two conclusions may then be drawn. First, the marginal rate of intertemporal product transformation, or the marginal internal rate of return on resources devoted to production, is equal to the market rate of interest and, second, the individual has maximized his attainable wealth measure by W* on the t0 axis. Given the endowment point E, it follows that by allocating EM = RQP0 to production, the next period's output can be increased by MP*=RXPX. The total resources available for consumption in the next period would then be the sum of 0/?i, the prospect of next period's income included in the present endowment, and R\P\, or OPi =PoP*. But even though the point P* represents the maximum attainable wealth position, it does not represent the individual's optimum intertemporal consumption allocation. Consider now the indifference curve labeled II, which cuts the production frontier and the valuation line at P*. The individual can attain a higher level of utility by moving back along the highest attainable valuation line in the direction of the arrow. In Figure 3.3, he will move to the optimumconsumption point C*, at which an indifference curve is tangential to the valuation line. At that point, the marginal rate of intertemporal resource substitution in consumption will be equal to the market rate of interest and also, therefore, to the marginal rate of intertemporal resource transformation in production. By taking these actions, the individual has made two moves that, by virtue of the given market rate of interest, we can regard as separable, thus laying the foundation for what we anticipated as a separation theorem. This states that the optimum intertemporal consumption decision is separable from the corresponding production decision. The individual can thus be thought of as moving first from his endowment point E to the optimum-production point P* and then back to the optimum-consumption point C*. In order to make the move from P* to C*, the individual will borrow at the market rate of interest an amount of consumable resources equal to C*N in present value terms, and he will therefore have to repay the equivalent of this amount out of the next

3 Concept and relevance of economic value

53

Figure 3.4 period's production, or the amount of P*N in next-period values. This borrowing and subsequent repayment transmutes the optimum-production vector (Po, Pi) into the optimum-consumption vector (Of, Cf). It permits the individual to achieve a consumption vector that lies outside of the production possibility frontier, PPr. Alternatively, by comparing the original endowment point E and the finalconsumption point C*, we could regard the overall operation as reducing the current consumption from the current endowment of Ro to Of, the difference between these two amounts being allocated to production. Then a further amount of C$P0 is borrowed and also allocated to production in order that in the next period the total amount of resources will be increased from the endowment of Ri to PoP*. Out of this latter total the amount of NP* will be used to repay borrowing, and the residue of PoN = 0Cf will be available for consumption. A similar condition, focusing on the financing of resource utilization and production in the firm, can be observed from Figure 3.4. Here the firm is assumed to have no initial resource endowment, and its intertemporal production possibility frontier, described by the concave locus OP in the left-hand quadrant of the figure, therefore emanates from the origin. Magnitudes to the right of the origin on the horizontal axis represent individuals' positiveresource availabilities measured in present values. Magnitudes to the left represent resource amounts borrowed by firms from individuals and employed as inputs to production. The concavity of the production possibility frontier in the left-hand quad-

54

I Theoretical issues and analytic motivation

rant of the figure indicates that the firm experiences diminishing marginal productivity as the level of resource input is increased. In the conditions envisioned, the firm will borrow an amount equal to 0R0 in the present period and will use this as an input to production to realize an output of RQP* = 0/?I in the next period. It is assumed that the firm will borrow the money capital needed to finance this production at an interest rate of r percent, given by the slope of the parallel valuation lines P*A and R\B. The optimality of the production point P*, where the production possibility frontier is again tangential to a valuation line, follows from the equality between the rate of interest at which it borrows and the marginal internal rate of return in production. The amount of borrowing equals OR0 = R\P* =AB, and the repayment out of the next period's production will equal R\D. This implies that of the next period's production of 0/?i, the net amount remaining to the owners of the firm will be OD. Alternatively, in current-value terms, the gross output from production equals 0Bf of which AB is used to repay borrowing and OA remains for the owners. The owners' intertemporal consumption opportunity set is described by the triangular space ODA because the future net income of OD has a present value of OA. The owners need not wait, however, until the acquisition of OD resources in the next period before they consume. Given the market rate of interest, they may move down along the valuation line DA until they reach their optimum consumption point C*. At that point their subjective marginal rate of substitution between present and future consumption, the slope of the indifference curve at C*, is equal to the market rate of interest at which they have borrowed. The owners, then, will borrow OCo = CiC* for present consumption, in exchange for which they will repay CXD out of their future expected net income. The remaining part of that net income, OCi, will then be available for consumption in the next period. In this exercise, given the market rate of interest, the firm's investment, production, and financing decisions are separable from the owners' consumption decision. The form of the owners' consumption utility function, in other words, is quite irrelevant to the firm's optimum borrowing, investment, and production decisions. Given the market rate of interest and given the ability of both the firm and its individual owners to borrow at that rate, the only thing the firm's managers need worry about is the maximization of the present value of the future expected revenue stream of the firm. If, in the example, the firm produces at a maximum point P*t the owners' ability to borrow at the market rate of interest, or, that is, to sell claims against their future expected income, enables them to achieve any intertemporal consumption vector they wish. The notion of economic value that we have developed in this chapter, along with the adoption of the maximization of economic value as the firm's over-

3 Concept and relevance of economic value

55

riding objective, is prominently displayed in the neoclassical theory of the firm. That body of theory has not yet, however, reached a completely satisfying integration of the valuation problem with those of the simultaneous optimization of the firm's production, capital investment, pricing, and financing decisions. In the chapters of Part II that follow, we shall investigate the main lines of the integrative structures that might be forged within the neoclassical theory. We shall be able, then, to set against this body of thought the newer perspectives we shall adduce in Part III.

PART II

The neoclassical tradition

CHAPTER 4

Production, pricing, investment, and financing interdependence in the firm The theory of the firm in a real-time, money-using economy must be grounded in the interdependent theories of production, capital investment, and finance. Traditionally, the theory of the firm has begun its analysis of production by positing a Marshallian short-run period in which, as the real capital employed in the firm was given and fixed, attention could focus on choosing the variable-factor inputs with which to produce specified levels of output. Then, at a later stage, the assumptions of the Marshallian long run took account of the possibility or desirability of varying the fixed factors also. The problem in this scheme of things is that no significant place exists for money capital. The real capital-money capital dichotomy is not addressed. No meaningful discussion is given of the need for money capital, the sources from which it might be obtained, and the manner in which the cost and availability of it constrain the firm's structural and operating decisions. The Marshallian short-run and long-run fictions did, of course, point in the direction of an accounting for real time, as did Marshall's concern for the evolution, growth, and decay of firms. But the analogy of the "trees of the forest" (Marshall, 1920, p. 315f.), together with that of the representative firm, effectively collapsed the real-time argument to a timeless form and prepared it for Pigou's notion of the optimum size of the firm and its equilibrium condition (see Robinson, 1969, pp. v-vi). Progress in analysis requires that priority be given to what we have called the ab initio planning model of the firm, so that money capital, with all of the questions of intertemporal valuation and uncertainty that it opens up, can be integrated immediately into the theory. We shall sketch in preliminary outline in this chapter the way in which that might most effectively be done, consistent with the thought forms of the neoclassical theory. It will not be possible to take full account initially of uncertainty or of the full significance of the passage of time. In this sense, the model is quite incomplete. We shall return to those matters. It should be emphasized, moreover, that the analysis that follows remains firmly within the neoclassical paradigm and that its principal significance lies in the manner in which it brings into focus the question of production-capital-financing interdependence. That will in due course be subject to an alternative interpretation in Part III, where the present neoclassical assumptions are amended. We are here, moreover, envisaging a static model of long-run structural 59

60

II The neoclassical tradition

optimization in the firm. The model is not at this stage dynamic, and the structural optima it brings into view are understood, as in the general neoclassical static theory, to be reproduced from period to period by the determinant relations in the model. The static nature of the argument implies that as the firm is not in this context growing over time, all of the net income is paid to the owners in the form of dividends. At this early stage of our work, all financing is assumed to be obtained from external sources, and the cost of money capital is correspondingly the cost of externally available funds. Notation employed in the model The geometry of the firm's revenue, production, cost, and profit functions will be familiar from earlier studies in the theory of the firm. Principal attention will therefore be given in what follows to a straightforward algebraic statement of the theory, without the familiar geometric support. Brief comments will be made on some of the principal components of the model after the following list of notation employed.1 p = equity owners' capitalization rate, or required rate of return, which is functionally dependent on the equity and debt capital mix employed in the firm /? = unit selling price of the firm's product (assuming a singleproduct firm), regarded as a function of the quantity of output produced and sold, Q, and written asp=p(Q), thereby recognizing the presence of imperfect competition in the firm's output market X, Y = input factors of production (confining attention initially to a two-factor production function as a means of illustrating the principal relations involved), to be described more precisely as the number of units of the factor capacities employed Q — the level of output, shown as dependent on the level and mix of factor inputs and described in the familiar production function form as Q =f(X, Y) 7i> 72 —unit factor costs of inputs X and Y, respectively, to be referred to more precisely as the periodic operating flow costs per unit of factor capacity input r = average rate of interest on the total debt or nonownership capital employed in the firm, understood, as in the case of 1

The model in this chapter is derived from that first introduced in Vickers (1968) and expanded in Vickers (1970). The argument has been reproduced in Turnovsky (1970) and has been discussed extensively in Herendeen (1975, p. 97f.).

4 Production, pricing, investment, and financing 61 the equity owners' required rate of return, p, to depend on the debt-to-equity ratio in the firm's financing structure K= amount of equity or ownership capital invested in the firm D = amount of debt or nonownership capital employed in the firm g = net working capital requirement function, describing the firm's net investment in working capital assets (cash, accounts receivable, and inventory, less current liabilities) as functionally dependent on the level of output, g = g ( 0 , or, indirectly, g = g[f(X,Y)] a, (3 = money capital requirement coefficients of factors X and Y, respectively X = an undetermined (Lagrangian) multiplier or coefficient attached to the money capital constraint variable

Preliminary comments on elements of the model 1

Costs of debt and equity capital

Both the equity owners' capitalization rate and the average rate of interest on the debt are dependent on the financing mix employed in the firm. They will be determined by the supply-and-demand conditions in the equity and debt capital sectors of the money capital market. This acknowledges the impact of the leverage characteristics of the firm's financing structure that we noted in Chapter 2. The higher the debt-to-equity ratio, or the higher the degree of financial leverage at work in the firm, the greater will be the risk exposure of both the creditors, that is the debt holders, and the residual owners. If we assume risk aversion on the supply side of the money capital market, the higher risks induced by higher degrees of leverage will call forth higher required rates of return on the invested money capital. This dependence will be exhibited by the functional forms r = r(K, D) and p = p(K, D). Both the equity cost and the debt cost functions will be understood to be monotonically positive. Increasing risks incur increasing money capital costs. At several points, it will be necessary not only to speak of the overall or average cost of money capital but also to consider the marginal cost of introducing additional capital to the firm. In the case of debt capital, for example, the "marginal direct cost of debt" will be defined as the rate of increase in the total interest burden of the firm that results from the employment of a marginal unit of debt capital. The interest burden will be defined as the total interest payments on the firm's debt capital, or rD, or, given the dependence of the average rate of interest, r, on the debt and equity financing mix, as r{Ky

62

II The neoclassical tradition

D)D. The marginal direct cost of debt follows as the derivative of this interest burden with respect to debt capital.

2

Revenue function

Observing the dependence of the unit selling price on the quantity of output sold, the firm's total revenue is described as R—p{Q)Q. Taking account of the dependence of output, Q, on the factors employed, the revenue function may be written as R =p(Q)f(X, Y). Again it will be necessary to envisage the marginal value of this function and to consider the marginal revenue obtainable from a variation in the firm's output and sales:

Or, in instances where it is desired to envisage the dependence of revenue on, say, the incremental employment of factor X, we may write

In this last expression, the notation fx refers to the marginal product of factor X, or the partial derivative of the production function with respect to factor X,d/(X, Y)/dX. Equation (4.3), therefore, describes the familiar marginal revenue product of factor X. It is equal to the marginal revenue multiplied by the marginal product of the factor. 3

Operating flow costs of factor inputs

The gamma terms in the list of notation refer to the periodic operating flow costs of the factors of production. Given that yi is the flow cost of factor X, the periodic cost of employing a designated number of units of that factor will be equal to yxX. If the unit flow cost should depend on the number of units of the factor being employed, the total flow cost of employing the factor during a given operating period would be j\(X)X. In the development of the theory, however, we shall not take a dependence of this kind explicitly into account. In the two-factor production function that we have assumed for purposes of exposition, the periodic production cost will be (4.4)

4 Production, pricing, investment, and financing

63

Difficulty attaches to the precise specification of these flow cost elements (see Vickers, 1968, p. 127f.). If the factor in view is a completely variable factor, for example labor, the operating flow cost may frequently be specified simply as the wage rate per unit of labor input. It is in that case precisely similar to the traditional concept of the unit factor price. It may be necessary, however, for the firm to invest in certain fixed assets to make the employment of labor possible, quite apart from the investment in real capital with which the labor cooperates in production. The operating flow cost of labor may therefore include certain maintenance, servicing, and depreciation costs on associated durable assets, in the same way as those charges affect the operating flow costs of the units of durable factor capacities. Let us observe initially, therefore, the manner in which, in general, the flow cost of a unit of durable factor capacity may be specified. The problem before us arises because the relevant asset life extends over more than one operating period. In order to maintain the presence of the asset in the firm, it will be necessary to incur in each period certain maintenance and servicing costs. At the same time, it will be necessary to charge against the income stream each period a contribution to the depreciation sinking fund that is being built up over the life of the asset to replace it when its economic life comes to an end. Only if this is done will it be possible to maintain the initial capital investment intact. This depreciation charge is based on the assumption that the asset will be regularly employed and fully used at its optimum technological intensity and will therefore be subject to regular wear and tear during its life. The operating flow cost will then be the sum of both these two cost elements, the maintenance and servicing cost plus the periodic depreciation. It will not include the interest payments on the money capital that has been raised to finance the investment in the asset. Those finance costs, which must be imputed to the factors of production whose employment the money capital makes possible, will be included as separate elements in the actual total flow costs of the factors. We shall return to that in a later section of this chapter. Let us assume that the firm contemplates investing in a physical asset capable of providing a stipulated number of ^-capacity units per operating period. With reference to this asset, we employ the following notation: L = economic life of the asset2 M = initial money capital outlay cost S = periodic maintenance and servicing cost 2

The choice of the optimum economic life of the asset gives rise to a suboptimization problem that is discussed at some length in Vickers (1968, Ch. 7). In the present introduction to the theory of the firm it is assumed that the optimum economic life of the asset is given.

64

II The neoclassical tradition T = periodic amortization or depreciation sinking fund installment r = rate of interest assumed for sinking fund computation

The interest rate r assumed for sinking fund purposes will be an approximation to the firm's lending rate if the sinking fund is invested outside of the firm. If, on the other hand, the sinking fund is invested in income-earning assets in the firm, and thereby serves as a source of money capital that substituted for new debt or equity security issues, the sinking fund rate of interest may be approximated by an estimate of the firm's cost of external money capital. Maintenance expenditures and replacement are two different ways of providing for the productive presence of the assets in the firm, and the same sinking fund r will be used in what follows in the consideration of the maintenance and servicing costs. The periodic operating flow cost of the units of F-factor capacity will then be the sum of the per unit values of T and S, or y2 = (T+S)/Y. The periodic maintenance and servicing cost can be expected to increase during the life of the asset and may be conceived, for purposes of example, to be equal to A dollars in the first period and grow at a rate of b percent per period. In any future period, the relevant cost will therefore be Aebt dollars. The total monetary outlay connected with maintaining the productive presence of the asset in the firm is then made up of two parts: (1) the succession of periodic costs equal to Aebt and (2) the provision of M dollars for replacement purposes at the end of the life of the asset. We abstract for the present from possible changes in price levels and assume that the scrap value of the asset at the end of its life is zero. Consider first the periodic depreciation charge. It is required that the replacement sinking fund amount to M at the end of L periods, or at the end of the life of the asset. It is therefore required that M = Jo Tert dt

(4.5)

Integrating Equation (4.5) and simplifying yields the result rM e -

(4.6)

rL

thus specifying the necessary periodic sinking fund contribution. Similarly, the periodic maintenance and servicing cost will be equal to the constant periodic S component of a stream of payments that has the same capitalized value as the actual stream of maintenance and servicing costs made necessary by the investment. The following equation may therefore be established:

f

f ~

r

t

dt

(4.7)

4 Production, pricing, investment, and financing

65

Integrating both sides of Equation (4.7) and solving for S yields o

rA b — r\

e'^ —

Taking the periodic T and S components, it follows that y2 = (T+S)/Y. There may not exist, of course, a single unique way of providing a stipulated number of units of factor Y capacity per period. It may be possible to provide the requisite capacity by investing in any of several different asset structures, each having its own technological characteristics. Different types of equipment may be employed having different prospective durabilities, initial capital outlay costs, maintenance and service charge characteristics, and economic lives. A schedule of economic characteristics of different alternatives may be drawn up, where Lif Mif Aif biy Tit Si, and yt represent the relevant data for the /th such alternative. For any given or desired level of factor Y capacity, then, the optimum real capital investment would be that which promises the minimum yt as thus ascertained. This magnitude should be incorporated in the planning stage in the functional relation y,(F), where the dependence of the operating flow cost on the level of the factor capacity employed is explicitly recognized. If the employment of such a variable factor as labor carried along with it the need to invest in certain durable asset facilities in order to make the employment of labor possible, such as transportation or catering equipment, it would be necessary to estimate the optimum T and S component costs of such assets in the same way as in the foregoing durable asset case. These would then be incorporated along with the wage rate, in the true operating flow cost of the labor units. 4

Money capital requirement coefficients

The a and /3 terms in the list of notation describe the money capital requirement coefficients of the respective factors X and Y. They indicate, for example, that for every unit of factor Y capacity employed in the firm it will be necessary to invest (3 dollars of money capital in fixed assets. The specification of these coefficients follows from the preceding discussion of the operating flow cost parameters. If, for example, it was found that the optimum method of providing for the presence in the firm of Y units of factor capacity was to acquire an asset with an initial money capital outlay value of M dollars, the asset investment per unit of Y capacity would be given as P = M/Y

(4.9)

Given the dependence of M, the money capital outlay, on the level of factor usage required, the general specification follows:

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II The neoclassical tradition (4.10)

The factor's money capital requirement coefficient is again functionally dependent on the level of factor usage envisaged. The total money capital requirements for investment in fixed assets generated by the decision to employ specifiable quantities of factors X and Y can be defined as the sum of a(X)X and P(Y)Y. 5

Net working capital asset requirements

The balance sheet described in Chapter 2 indicates that in order to maintain the firm in continuous operation it will be necessary to invest part of the available money capital in cash balances, accounts receivable, and inventories of various kinds. Part of the financing sources, or part of the money capital required for this purpose, will be obtained from the current liabilities, which are also described on the balance sheet. Apart from the short-term loans from financial institutions such as the banks, funds may be made available by deferring payment of the firm's liabilities for inventory purchases, giving rise to a current liability for the corresponding amount. Taking the balance sheet total of such current liabilities from the total of current assets provides an indication of the net working capital requirements of the firm. Alternatively, the working capital requirements of the firm can be described as that portion of the current assets that is financed from long-term money capital sources. For to the extent that, as is generally the case, the current assets exceed in total the current liabilities, the latter are covering only a portion of the currentasset requirements. The remainder, therefore, must be financed from the longterm sources, long-term debt or equity capital. The reason why working capital is obtained from long-term financing sources, or why, in other words current assets normally exceed current liabilities, is that the firm thereby maintains a liquidity cushion that is deemed to be necessary to support its ongoing operations. In general, current assets are liquid in the sense that they can be turned into cash either immediately or during the firm's normal operating cycle. The current liabilities are liquid in the sense that they become due for payment at more or less frequent intervals throughout the same operating cycle. We therefore describe a net working capital requirement function, dependent on the level of the firm's production and sales, as g(Q). It will be useful in what follows to envisage such a working capital requirement function as W = g[f(X,Y)]

(4.11)

showing the dependence of such money capital requirements on the firm's basic factor employment decisions.

4 Production, pricing, investment, and financing 6

67

Money capital requirement function

Making use of the variables and parameters described in the foregoing, it is possible to bring the several strands of analysis together and describe the firm's money capital requirement function. Using the symbol MCR to refer to the total money capital requirements, it follows that MCR = g[f(X, Y)] + a(X)X + P(Y)Y 7

(4.12)

Money capital availability constraint

At the outset of the firm's ab initio planning, it is not known what levels of factor X and Y employment will be optimal. But whatever level of production and factor use is decided upon, a total money capital requirement as defined in Equation (4.12) will be generated. Whatever level of production is chosen, it must be one that does not generate a money capital requirement greater than the money capital actually available to the firm. Defining the equity and debt capital available to the firm as K and D, respectively, the following relation must be satisfied: g[f(X, Y)] + aX + pY

(4.38)

Leaving Equation (4.38) in the form indicated for the present, consider the properties of the money capital availability constraint. It follows that when the optimization conditions are satisfied and the money capital availability constraint is operative, (4.39) Differentiating this equation throughout with respect to K yields

j||

4jL=i

(4.40)

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II The neoclassical tradition

From Equation (4.38) and the substitution of Equation (4.40), dV/dK=X

(4.41)

If X may thus be treated as an approximation to the marginal value productivity of equity capital, reliance may be placed on the solution value of X as an indication that the optimum amount of equity capital, and thus the overall optimum structure of the firm, is being approached. If, for example, the optimum structure of the firm is determined, subject to the constraint of a given amount of equity capital K and the solution value of X is greater than unity, the signal is thereby being given that the economic value of the owners' investment, V, could be increased by more than one dollar by the introduction of another dollar of equity, K. When the equity investment, along with the increase in debt capital optimally associated with it, has been increased so far that the solution value of X is unity, no further marginal investment of equity is desirable (apart from the modification introduced by the indirect effects we shall consider in the following section). In this way, and on the level of approximation we are at present examining, reliance may be placed on the solution value of X as the optimization signal when, as before, structural optima are determined in the manner specified in Equation (4.28). It follows from Equation (4.31) that when the solution value of X has been depressed to unity,

r=r+Dm+vH>

(4 42)

-

Thus, when the introduction of debt capital, together with successively larger commitments of equity capital, has been carried far enough to depress the solution value of X to unity, the full marginal cost of borrowing, shown on the right-hand side of Equation (4.42), will have been brought into equality with the solution value of p, the owners' capitalization rate or required rate of return, or the cost of equity capital. Further comment can be made on the relation between these money capital costs at infra-optimum stages or levels of planning. We write Equation (4.31) in the form

( £ & )

(4 43)

-

At infra-optimum levels, when the solution value of X is greater than unity, debt capital may be introduced even though the full marginal cost of borrowing, described in the parentheses on the right-hand side of Equation (4.43), is greater than the cost of equity capital, p. The explanation of this paradoxical statement resides, of course, in the fact that in the conditions envisaged, the size and structure of the firm is such that the marginal productivity of money

4 Production, pricing, investment, and financing 79 capital is still relatively high. This is indicated by the relatively high solution value of X. Thus a high marginal cost can be paid for debt capital so long as the marginal productivity of money capital in the firm is high enough to afford some benefit to the owners, or at least high enough to cover the full marginal cost of borrowing. For the latter takes account of the tendency for the increase in debt to raise the cost of equity [note the term dp/dD in Equation (4.43)]. Further interpretation may be assisted by reference to Equations (4.35) and (4.36). The former may be reinterpreted as Cx = Mxlp\

(4.44)

Equation (4.44) indicates that at infra-optimum levels of planning, the surplus marginal revenue product of additional factor employments, discounted at a discount factor equal to pX, should equal the marginal money capital outlay required by that factor employment. But at such infra-optimum stages, the implicit discount factor indicated is greater than the cost of equity capital, the amount of the difference being dependent on the value of X. When the solution value of X is unity, the discount factor, pX, will be reduced to p. A further approximation to money capital optimization To this point, however, the analysis has not taken full account of the beneficial effects that can be expected to follow, on both the debt and the equity costs, from the introduction of marginal equity capital. Additional equity capital will reduce the degree of financial leverage in the firm and therefore the risk exposure of the debt and the equity holders. The necessary further interpretation can now be made most effectively by returning to Equation (4.28) and differentiating partially with respect to K, setting the partial derivative equal to 1. This is done because at the optimization margin an extra dollar of equity capital must generate at least one additional dollar of gross economic value for the owners: >

=

\(w

dp ^^ +D

—VR W

dr\, , +x= l

.

/A

(445)

...

It follows from a rearrangement of Equation (4.45) that under full optimization conditions,

If, now, Equations (4.43) and (4.46) are compared and rearranged, it follows that under full optimization conditions

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II The neoclassical tradition

Thus the discount factor at which the marginal income streams should be discounted for planning purposes, pX, is equal, at the overall optimum structure, to what can be referred to as the full marginal cost of debt and the full marginal cost of equity. Equation (4.47) implies that this discount factor, pX, will, in the general case of risk aversion on the part of the suppliers of money capital, be less than p and greater than r. That is, r. Imagine now that a new variable 5 is defined as the sum of the simultaneous observations of these X and Y variables. We must interpret S as a random variable because any variable that is a function of one or more random variables is itself a random variable. We now write S=X + Y

(5.14)

It can be shown that E(S) = E(X) + E(Y)

(5.15)

leading to: Proposition 1 The expected value of a sum of random variables is equal to the sum of the expected values of the random variables. Interest frequently attaches to the weighted sum of random variables, where, for example, denoting the weighted sum as 5, S = wlX + w2Y

(5.16)

where w\ and w2 are the weights attached to the random variables. It then follows that E(S) = wxE(X) + w2E(Y)

(5.17)

leading to: Proposition 2 The expected value of a weighted sum of random variables is equal to the weighted sum of the expected values of the random variables.

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II The neoclassical tradition

The computation of the variance of a sum of random variables is a little more complicated. Using the expected-value notation, and writing the variance of a random variable as Var(X) = E[[X- E(X)]2]

(5.18)

the variance of the sum of the random variables X and Y, Var (S), can be written as Var(S) =E[[S-E(S)]2]

(5.19)

Var(S) =E[[(X + Y) - [E(X) + E(Y)]} 2 =E[{[X-E(X)] + [y-£(y)]j ]

(5.20)

or

We now let [X-E(X)] = dx, or the deviation of the X magnitudes from their expected values, and similarly let [Y — E(Y)] = dy. Equation (5.20) can then be written as Var(S)= E[[dx + dy]2]

(5.21)

Recalling the elementary proposition that

we can apply this formula on the right-hand side of Equation (5.21) and write the resulting expression for Yax(S) as Var(S) = E[d2 + d2 + Id^y]

(5.22)

Placing the expectation operator inside the brackets in Equation (5.22) in accordance with Proposition 1, the expression for the variance becomes Var(S) = E(dl) + E( d2) + 2E{d4y)

(5.23)

We recognize that the expectation terms on the right-hand side of Equation (5.23) describe, respectively, the variance of X, the variance of Y, and the covariance between X and Y. The final expression for the variance of the sum of the random variables is therefore Var(S) = Var(X) + Var(y) + 2 Cov(X, Y)

(5.24)

We can therefore state: Proposition 3 The variance of the sum of a pair of random variables is equal to the sum of the variances of the variables plus twice the covariance between them.

5 Probability, risk, and economic decisions 95 The expected value of the sum of the random variables, as defined, for example, in Equation (5.15), takes no account at all, and is in fact quite independent of, whatever covariation might exist between the random variables. But the covariance between the variables, as in Equation (5.24), enters in an important way into the determination of the variance of the sum. Much depends, therefore, on the magnitude of the covariance and, in particular, on the sign of the covariance term. We can see the impact of this in another way. Making use of Equation (5.13), we write Var(S) = Var(X) + Var(y) + 2 pXY(ix g(Uo) whenever U\>UQ. Then the utility function T also represents the preference ordering. It follows that if, working with the original utility function, a number of 60 is assigned to one combination of X and Y and a number of 30 is assigned to a second such combination, it is not possible to say that the first combination provides twice as much utility as the second. In order to be able to say, moreover, that a preference ordering can be represented by an ordinal utility function, or that it is, in formal terms, a representable preference ordering, a number of characteristics of the ordering must be present. First, in addition to the notions of completeness and transitivity, the set of possible objects of choice must be "connected." That is to say, if A\ and A2 are combinations of commodities in the opportunity set available to the individual, it must be possible to proceed from Ai to A 2 along a continuous path of such combinations. This important assumption of continuity will reappear in a significant guise in the theory of preference orderings over stochastic outcomes. Second, we can consider all the combinations of commodities in the closed opportunity set that are at least as well liked as a given combination, say Ai, and also the set of combinations that are preferred to A\. This means that if we envisaged an infinite sequence of commodity combinations in, say, the set of combinations preferred to A\ and such a sequence converged to a limiting combination of Ao, that limiting combination Ao would also be preferred to A\. Let us consider an ordinal utility function

110

II The neoclassical tradition

U=f(X,Y)

(6.1)

It may appear that we could envisage the increment of utility that would result from adding a marginal increment of, say, commodity X to the consumption combination. It might be thought that in this way, assuming the utility function possesses appropriate differentiable properties, we could define the partial derivative of the utility function with respect to X as the marginal utility of X. But difficulties attach to such an interpretation of the ordinal and nonunique utility function we have just described. We shall be interested, however, in the ratio of the marginal utilities of the commodities, and this, which is quite properly reflected in the ratio of the partial derivatives of the utility function, has valid and significant meaning. Such a ratio remains defined over a monotonically increasing transformation of the original utility function. We may take, for example, the function U = aX + bY and a monotonic transformation of it, T=U2. The function T can then be written T=a2X2 + b2Y2 + labXY. The ratio of the partial derivatives of the function U can be shown to be equal to alb. When the similar ratio of partial derivatives is calculated from the function T, it is found to be equal once again to alb. This ratio of partial derivatives will be referred to as the marginal rate of substitution between the commodities, or between whatever happens to be defined as the objects of choice that form the arguments of the utility function. At this point, the sole property of the consumer utility function in which we are interested is the manner in which it gives rise to a commodity indifference map. This map is made up of a family of indifference curves in the commodity space such that each curve in the family describes a set of commodity combinations that afford the individual the same level of utility. In that sense, he is indifferent between them. They rank equally in his representable preference ordering. Members of such a family of indifference curves are shown in Figure 6.1. The marginal rate of substitution between the arguments in the utility function, in this case commodities, can be developed in the following manner. Consider the utility function in Equation (6.1). We are interested in the change in the value of the function that would result from a simultaneous change in the magnitude of the arguments X and Y. We specify this by writing the total differential of the function: dU = ^dX

+ ^dY

(6.2)

Notationally, dU/dX and dU/dY in Equation (6.2) may be referred to as/* and fy, respectively. The overall change in the value of the utility function is defined as the rate of change with respect to commodity X, or the partial derivative with respect to X, multiplied by the amount of the change in X, that

6 Utility, uncertainty, and theory of choice

111

Figure 6.1 is, dX, plus the rate of change with respect to Y multiplied by the amount of the change in Y. We consider now a simultaneous change in the arguments X and Y that leaves the individual on the same indifference curve, or at the same level of utility. In that case we can set Equation (6.2) equal to zero. It follows by transposition that dY dX'

fy

(6.3)

The derivative dYldX shown in Equation (6.3) describes the slope of the indifference curve in Figure 6.1. We conclude, therefore, that the slope of the indifference curve is defined as the negative of the ratio of the partial derivatives of the utility function, or as the ratio of the marginal utilities. It defines the marginal rate of substitution between the commodities X and Y, or the rate at which, in order to remain at the same level of utility, the individual is willing to substitute commodity X for commodity Y. The consumer's utility maximization problem may be solved against an income constraint that restricts him to an attainable commodity combination, or to an attainable commodity set such as the area OAB in Figure 6.1. At the optimum-commodity combination point, or at the point of constrained utility maximization, the slope of the indifference curve will be equal to the slope of the relative commodity price line AB that forms the boundary of the attainable

112

II The neoclassical tradition

set in the commodity space. The location of that boundary is set by the total income or resources available for consumption. The fact that the budget constraint is operative ensures that the two equal slopes meet in a tangency. This tangency condition is shown in Figure 6.1 at point D. The individual will consume X* of commodity X and 7* of commodity Y. An analogous tangency condition will occur in the case of a portfolio investor's optimization over a stochastic utility function. Utility and probability: investor's risk-aversion utility function Consider now the investment opportunities and prospects confronting an investor who approaches the financial asset market at the beginning of time period t with a specified total wealth endowment of Wt. We can conceive of the rate of return, Rt9 that might be earned on that wealth portfolio during period t. In the manner of Chapter 5, we can visualize that rate of return as a random variable describable by a subjectively assigned probability distribution, such as is shown in Figure 6.2, with expected value E(Rt) and standard deviation (jiRt). The probability distribution of the rate of return might or might not be the approximation to the normal distribution depicted in Figure 6.2. If the distribution is assumed to be normal, the third moment, or the measure of skewness, will be equal to zero, and the investor's attention will be focused on the first two moments, the expected value and the standard deviation. This is the usual form of distribution assumed in the traditional theory, which is accordingly referred to as the mean-variance theory of investor's portfolio optimization. Consider now the potential accumulated value of the investor's wealth at the end of time period t: Wt+l = (l+Rt)Wt

(6.4)

As the rate of return on the portfolio is a random variable, we must interpret the end-of-period wealth, Wt+\, as a random variable also. We are interested, then, in the attractiveness to the investor of, or the different possible levels of utility provided by, the possible values of Wt+i. It follows that we can focus on the properties of the random rate of return that generates the stochastic nature of the wealth variable. For purposes of exposition we shall assume an ordinal utility function defined over this random rate of return, and for concreteness we shall assume it to be a second-degree utility function of the form U(R) = aR-bR2

(6.5)

This function specifies the level of utility that the individual would attach to different outcome values of the rate-of-return variable. As utility is now a function of a random variable, however, utility itself must be interpreted as a

6 Utility, uncertainty, and theory of choice

113

Figure 6.2

random variable. This implies that Equation (6.5) does not possess differentiate properties. No meaning attaches to the derivative of a random variable with respect to another random variable. No meaning would attach, therefore, to the notion that the marginal utility of the rate of return might be described by the derivative of the utility function with respect to the return variable. In order to address the concept of marginal utility, the utility function in Equation (6.5) must be transformed into another function that does have differentiate properties. We accomplish this transformation by taking the expected values of both sides of Equation (6.5). The development in Chapter 5 implies that E[U(R)] = aE(R)-bE(R2)

(6.6)

The second term on the right-hand side of Equation (6.6), the expected value of the square of the underlying random variable, can be shown to be equal to the sum of (i) the square of the expected value and (ii) the variance of the probability distribution of the random variable. That is, This can be confirmed from an inspection of the probability distribution in Figure 6.2. Consider the possible outcome recorded in Figure 6.2 as Rt. This deviates from the expected value by the amount dt. It can therefore be written as Rt = E(R) + dt. Taking the square of this magnitude provides (6.8)

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II The neoclassical tradition

Equation (6.6), however, requires us to evaluate the expected value of R2. We therefore take the expected value of Equation (6.8). In doing so, we interpret the expected values of the terms on the right-hand side of the equation in the following manner. The expected value of the first term, the expected value of the square of the expected value of the distribution, is simply the square of the distribution's expected value. This is because that expected value is a datum, or a constant, once the form of the underlying probability distribution has been specified as in Figure 6.2. The expected value of the second term, or E(dj), is recognizable as the variance of the probability distribution. Finally, the expected value of the third term, or 2E(R)E(di), is equal to zero because the last element in this expression is simply the weighted sum of the deviations of the possible values of the variable from their average, or expected, value weighted by their associated probabilities. We therefore write the expected value of Equation (6.8) in the form of Equation (6.7) as required. Taking Equation (6.7) and substituting into Equation (6.6) and writing the expected value of the probability distribution of the rate of return as /UL and the variance as a2, the expected utility function can be written as E(U) = aim- b\x2 - be2

(6.9)

It is important to note what has been accomplished by this transformation. We have taken a second-degree utility function defined over the random variable rate of return and have transformed that into an expected utility function defined over the first two moments of the probability distribution of the underlying random variable. If there had been reason to believe that the probability distribution of the rate of return on the portfolio possessed a positive measure of skewness, and if, furthermore, the investor was attracted to the possibility of the high rate of return that such a positive skewness indicated, we could have begun with a third-degree utility function defined over the random variable. Such a third-degree stochastic utility function would be transformed into an expected utility function defined over the first three moments of the distribution of the underlying random variable. Generalizing from this result, an nth-degree stochastic utility function can be transformed into an expected utility function defined over the first n moments of the probability distribution of the underlying random variable over which the stochastic utility function is defined. If, then, the investor is optimizing over a second-degree, or quadratic, utility function, there is no point in examining the third or higher moments of the relevant probability distribution. Similarly, if the probability distribution is assumed to be a two-parameter distribution, such as the widely assumed normal distribution of economic and financial data, there is no point in assuming that the utility function is of higher than second degree. This provides the basis for the mean-variance theory of portfolio selection and optimization.

6 Utility, uncertainty, and theory of choice

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Given the expected utility function of Equation (6.9) and recognizing its differentiable properties, we can envisage the marginal utility of expected return by taking the partial derivative with respect to the expected-return variable:

*fpi = .-2^

(6.10)

Similarly, the marginal utility of risk, or the partial derivative with respect to or, provides

m

(6.11)

da

These results indicate that the marginal utility of risk, as indicated in Equation (6.11), is everywhere negative. We say, therefore, that an individual optimizing against a utility function of this form is a consistent risk averter, or that the utility function is a consistent risk-aversion function. Other functional forms and measures of risk aversion have been defined in the extensive literature on utility functions. The illustrative form with which we have worked is frequently referred to as a Tobin function (Tobin, 1958). It would be possible to adopt an exponential utility function of the form U = a-ce~bR

a^0,c,b>0

(6.12)

which transforms into an expected utility function (see Ford, 1983, pp. 23-4, 191-92) defined as E(U) = a-ce~b^R-(b/2)(rR]

(6.13)

An interesting difference exists between the Tobin function and that in Equation (6.12). In the former case, it appears from Equation (6.10) that the marginal utility of expected return will become negative for values of expected rate of return greater than allb. In the latter case, the marginal utility of expected return is everywhere positive, and the E(U) function has an upper bound of a. It is not necessary for analytical purposes, therefore, to place a limit on /UL to guarantee that the marginal utility is positive as in the case of the quadratic utility function. If it can be assumed that the marginal utility of expected return is positive, and if the utility function is a consistent risk-aversion utility function where the marginal utility of risk is everywhere negative, as in Equation (6.11), the iso-utility or indifference contours in the risk-return plane will be positively inclined as in Figure 6.3. As in the analysis of the consumer utility function, the slope of the indifference curve is defined by the negative of the ratio of the marginal utilities or the partial derivatives of the function with respect to

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II The neoclassical tradition

o(R)

Figure 6.3 its arguments. Making use of Equations (6.10) and (6.11), this slope is defined as d/JL _

da

2bcr

a — 2b JX

(6.14)

In Figure 6.3, the arrow indicates the direction of increasing utility. The convexity of the indifference curves can be established by taking the second derivative d2/ui/dcr2, which may be shown to be positive in the present example. This analysis provides the basis for a choice between alternative portfolios depending on the probability distributions of their rates of return. If the expected values and standard deviations of portfolio rates of return are substituted into the expected utility function, the relative attractiveness of portfolios can be compared and evaluated. That portfolio will be the most attractive that offers the investor the prospect of achieving the highest attainable indifference curve. It is for this reason that decisions under uncertainty, or what we are here considering as probabilistically reducible risk, have been referred to as 4 'choices among probability distributions" (Mossin, 1973; Hirshleifer and Riley, 1979). Different possible degrees of risk aversion are exhibited in Figure 6.4. The indifference curves A and B shown there are derived from different utility functions, and each is a member of a different indifference map. It is imagined that the investor occupies the position marked P, holding a portfolio that promises an expected return of E{RP) and has a standard deviation of return of cr{Rp). Moving along the risk axis to the right of P, it is readily seen that a

6 Utility, uncertainty, and theory of choice

117

E(R)=L B

| Aa o(R)

Figure 6.4 higher increment of expected return would be required if the investor were optimizing over a utility function that generated indifference curve B than if his utility function had generated indifference curve A. The utility function that gives rise to indifference curve B would therefore exhibit the greater degree of risk aversion. When it is said that an investor is risk averse, this implies that his attitude to risk is such that he will reject what has been called a fair gamble. This refers to a situation in which the expected value of the outcomes of a risky prospect is equal to the amount that would have to be paid for the opportunity to acquire that prospect. For example, a lottery that offered two alternative outcomes of two dollars and zero dollars, where each outcome had a 0.50 probability of occurring, would have an expected value of one dollar. A "fair gamble" would be the exchange of the lottery prospect for its expected value of one dollar. The investor would stand to realize a net gain of one dollar if the two-dollar outcome occurred, or a net loss of one dollar in the event of the alternative outcome. But a risk-averse investor would reject such an exchange because the disutility to him of the loss would be greater than the utility he would obtain from an equivalent gain. His utility function, such as that assumed in Equation (6.5), would therefore be concave. An alternative measure of risk aversion has been constructed by focusing on what has been called the Absolute Risk-Aversion (ARA) or the Relative Risk-Aversion (RRA) properties of the utility function. A decreasing absolute risk aversion is said to exist if an investor is prepared to increase the amount of his wealth invested in risky assets when the total amount of his wealth increases. The opposite behavior would exhibit increasing absolute risk aver-

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II The neoclassical tradition

sion. Take, for example, a utility function defined over total wealth of the form U(W) = W-cW2

(6.15)

and employ the notation U'{W) and U'\W) to refer to the first and second derivatives of that function as defined over actual wealth outcomes. Then (see Elton and Gruber, 1984, p. 21 If.; Pratt, 1964; Arrow, 1971) absolute risk aversion may be measured by

U(W)

ARA= Jt(^}

(6.16)

In the case of the second-degree utility function assumed in Equation (6.15), the ratio defined in Equation (6.16) would be

The direction of change of this ARA as wealth increases, or d(ARA)/dW, can be shown to be J(ARA)

4c2

>{)

(618)

The function therefore exhibits increasing absolute risk aversion. Similarly, relative risk aversion has been defined with reference to the proportion of a wealth portfolio that is invested in risky assets as the total level of wealth increases. This has been defined as (6.19) An investor who places a greater percentage of his wealth in risky investments as his total wealth increases is said to exhibit decreasing relative risk aversion. By following the same procedure as before, it can be shown that the illustrative wealth utility function in Equation (6.15) exhibits increasing relative risk aversion. By analogy, a second-degree, or quadratic, utility function such as we assumed in Equation (6.5), or the so-called Tobin function, exhibits the properties of increasing absolute and relative risk aversion. This, it should be noted, casts some doubt on the relevance in actual fact of the widespread use of this form of utility function in the literature on financial asset optimization. It would seem that contrary to the logical implications that we have just deduced, investors in general do in fact place larger amounts of their wealth in risky assets as their wealth increases, or, in other words, they exhibit decreasing absolute risk aversion. Less certainty attaches to the behavior in actual

6 Utility, uncertainty, and theory of choice 119 fact of relative risk aversion. But the quadratic utility functional form has been extensively used, and risky asset choice theory has been substantially based on the implicit mean-variance assumptions. Moreover, it has been shown that quadratic functional forms can be found that do exhibit more desirable risk-aversion properties (see Mossin, 1973). Axiomatic basis of stochastic utility The expected utility theory we have examined is derived from the work of von Neumann and Morgenstern, whose propositions regarding the axioms of choice in risky situations have influenced the vast literature that has developed in this field (von Neumann and Morgenstern, 1953).l In what follows, we shall specify the axioms by making direct reference to the selection of portfolios of risky assets. A particular portfolio, designated as Piy will be described by a set of possible rate-of-return outcomes, associated with each of which is the probability of its occurrence. In Equation (6.20), the /th portfolio available to the investor is envisaged as promising n possible rates of return, /?i through Rn, with associated probabilities px through pn. In the following discussion, the uppercase Pt refers consistently to a possible portfolio and the lowercase pt refers to a probability magnitude. Pi=[(Ri,Pi),

(R2,P2), . . . , (Rn, Pn)]

Pi^O, 2>Pj indicates that portfolio Pt is preferred to portfolio Pj, and Pt~Pj means that the individual whose preference ordering is being considered is indifferent between portfolios Pz and Pj (see also Luce and Raiffa, 1957, Ch. 2; Hey, 1979). Axiom 1. Axiom of completeness or comparability Either Rt ^ Rj or Rj >: Rt for all /, j , = 1 , . . . , « . This axiom, which might be referred to also as the axiom of pairwise comparison, orders all possible pairs of the underlying rates of return on a portfolio. It states that the investor is able to describe a preference ordering over portfolio rates of return, such that his preference or indifference between all possible pairs of such rates of return can be established. Axiom 2. Axiom of transitivity of portfolio choices If Pi ^ Pj and Pj £ Pkt then Pt £ Pk. 1

The development given here follows that in Ford (1983, p. 18f.). A somewhat condensed treatment of the same subject is contained in Henderson and Quandt (1971, p. 42f.).

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II The neoclassical tradition

This axiom states, for example, that if the ith portfolio is preferred to the 7th portfolio, by virtue of its risk-return profile, and the7th portfolio is preferred to the Ath, then the Ufa portfolio will be preferred to the kth. These relations are assumed to hold for all pairs of portfolios in the set of portfolios available to the investor. Axiom 3. Axiom of continuity Assume that R\ is regarded as the most preferred rate-of-return outcome and that Rn is the least preferred. Then there exists a probability number, pit where 0 < / ? / < l , such that

Ri~l(Ri,pd, (Rn, 1 -pd]

for all 1= 1, . . . , n

This seemingly complex axiom will play a crucial part in the following construction of a utility function. It states that for each possible rate-of-return outcome Rt that may be designated (here shown on the left of the indifference sign), a combination of outcomes, and thus a portfolio choice, can be found that is indifferent to Rf. That portfolio would be made up of a combination of the most preferred and the least preferred outcomes, Rx and Rn, as shown on the right-hand side of the indifference sign. That is, there is some probability that could be assigned to the assumedly best possible outcome, with the magnitude of 1 minus that probability assigned to the worst possible outcome, that would make the individual indifferent between a combination of those best and worst outcomes and the designated outcome Rt. The magnitude of the probability pt that would have to be assigned for this purpose is dependent on the individual's attitude to risk and is therefore closely associated with the degree of risk aversion in his utility function. Axiom 4. Axiom of substitutability of outcomes and portfolios Let Rt and Qt be possible rates of return on a portfolio. If/?,-—Q,-, then Pi ~ P 2 , where l), Pl=

l(Rl,Pl),

( « 2 , / > 2 ) , • • • , (RhPi), (Rl,P2),

• • • , (Qi,Pi),

• • • > . . .

(Rn,Pn)] (Rn,Pn)]

This axiom makes a statement about the investor's attitude to two different portfolios, Pi and P2. The portfolios are understood to be the same in every respect except one. Portfolio Px contains a possible rate of return Rit whereas P2 contains a possible rate of return Qt. All other rates of return, and also their associated probabilities, are the same for both portfolios. If the investor is indifferent between the R; and the Qt rates of return, and if their probabilities of occurrence are the same for both portfolios, the investor will be indif-

6 Utility, uncertainty, and theory of choice

121

ferent between the two portfolios. This axiom is also referred to as the axiom of independence of irrelevant alternatives. Axiom 5. Axiom of monotonicity If, given Axiom 1, Rt ^ Rj and if

Pi = [(Rhpo)ARj, l-/?o)],and P2=[(Ri,Pi)ARj, l-/?i)],then Pi ^ P2

if and only if

/?o—Pi

This axiom states that an individual who compares two portfolios containing the same possible outcomes will choose that portfolio in which the preferred outcome has the higher probability of occurring. Axiom 6. Axiom of complexity IfPj=[(RhPij),

i = l , . . . ,n]

forj=l,

. . .,m

(that is, we suppose there exist m portfolios, Pp each of which contains n possible outcomes, Rh with associated probabilities/?/), and if (that is, there exists a "lottery" Si such that one of the portfolios Pj will be the prize in the lottery, with probability sj), and if (that is, there exists a portfolio S2 that contains n possible outcomes R[ with probability/?!), and if m

Pi= 2 ptjSj for i = 1, . . . , n, 7=1

then S\ This complex axiom is stating that probabilistic opportunities can be combined, depending on the probabilities specified, in such a way that an individual will be indifferent between (i) a given portfolio and (ii) a lottery in which one of the set of portfolios will be obtained as a prize. Interpretation of this rather complex case is assisted by focusing on the implicit probabilities of realizing the different possible portfolio rates of return by choosing (i) what we might call the direct route of S2 above or (ii) the indirect route of the lottery

Si. If the direct route of S2 is chosen, the probability of realizing a designated rate of return, Rh is, by definition above, pif which is equal, again by defini-

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II The neoclassical tradition

tion, to ^PijSj. Consider now the probability of realizing that same rate of return, Rh if the indirect route of S\ is chosen. In that case, we must consider the set of conditional probabilities involved. The first possible way of realizing R( is to obtain portfolio 1, Pu as the outcome of the lottery and then to realize Rt as the outcome of that portfolio. The probability of realizing Rt by this route, then, is the product of the probability of receiving Px from the lottery and the probability of Rt occurring conditional on the portfolio P\ having been obtained. A similar statement could be made regarding the probability of realizing /?, following the indirect result of each of the other possible lottery outcomes. The actual resulting probability of realizing /?, via the indirect lottery route will then be the sum of all of the products of probabilities thus obtained. We can write, therefore, piRf. S2) ~Pi— 2 Pusjas defining the probability J;= l of Rt via the direct route, and i = l , . . . ,n,j=l,

=p(Pi)p(Ri:Pi)+p(P2)p(Ri:P2)+ . . + Smpim

. . . ,m

•• •

jj

7=1

thereby defining the probability of/?/ via the indirect route. This, therefore, establishes the equality between the probabilities of portfolio outcome /?,- by either the direct or the indirect routes. The meaning of Axiom 6, on the basis of the assumptions specified, is accordingly established. Construction of the utility function In the following section we shall provide a detailed example of the construction of a utility function based on behavior in accordance with the foregoing axioms. An individual will be assumed to be acting rationally when making a choice between probability distributions, or between portfolios Pit if he acts in accordance with the axioms. This will imply that the individual will act in such a way as to maximize an ordinal utility function, where the utility of a portfolio can be written as U(P) and where that utility is a weighted sum of the utilities attached to each of the possible rate-of-return outcomes contained in the portfolio. These utilities of rates of return are weighted by the probability of their realization. We can therefore write U(P)= ^PiUi 1= 1

where ui=U(Rl).

(6.21)

6 Utility, uncertainty, and theory of choice

123

Let us lay the background for that discussion as follows. We now take the least preferred and the most preferred rate-of-return outcomes and assign to them arbitrary numbers on an ordinal utility scale over which it is desired to define the utility function. We may arbitrarily assign the utility value 0 to the least preferred outcome and the utility value 1 to the most preferred, designating these outcomes Rn and Rl9 respectively, as in Axiom 3. We now wish to find a way to allocate a number on the utility scale to any possible outcome between these least preferred and most preferred outcomes. Let such an outcome be designated Rt. Then we are assured by the axiom of continuity, Axiom 3, that it is possible to discover a probability magnitude such that Ri-KRupdARn,

I-Pd]

(6.22)

In accordance with the axioms, the individual is indifferent between the designated outcome Rif if it were available to him with certainty, and the chance of receiving either Rn or R\ with probabilities 1 —pt and ph respectively, as on the right-hand side of Equation (6.22). Both of these options must therefore be assigned the same number on the utility scale. Suppose that in relation to Rt the probability magnitude that established the indifference in Equation (6.22) was 0.75. Then we can substitute this probability magnitude into the right-hand side of Equation (6.22) and calculate the expected utility of the distribution of outcomes described there. That expected utility will be equal to the sum of (i) the utility that has been assigned to the most preferred outcome, namely 1, times the probability of realizing that outcome, namely 0.75, and (ii) the utility assigned to the least preferred outcome, namely 0, times the probability of realizing that outcome, namely 0.25. The expected utility calculates to 0.75, and that, therefore, is the utility magnitude to be assigned to the designated outcome of/?/. In a similar way, utility magnitudes could be calculated, relying repeatedly on the axiom of continuity, for assignment to all possible values of outcomes between the least and the most preferred. In this way, a utility function will have been defined over the entire range of possible outcomes. Once we have a utility function defined over the range of possible outcomes, it follows that the expected utility of a portfolio whose possible outcomes are described by a given probability distribution will be equal to the weighted sum of the utilities of each of the possible outcomes of the portfolio, where each such possible utility is weighted by the probability of its being realized. In other words, we have U(P)= I.piUiRd

(6.23)

i= 1

The investor, therefore, should select that portfolio that maximizes his expected utility in this sense.

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II The neoclassical tradition

The utility function, of course, must be consistent with the preference axioms. That is, we require that U(P i) > U(P2)

if and only if P { > P2

(6.24)

It can be shown from the axioms stated that this condition will be satisfied. Example of a derived utility function In the interest of concreteness, consider the following example of the derivation of a utility function defined over a range of possible outcomes. To change the area of application, let us imagine that a firm faces the prospect of a cash flow that may range between -$10,000 and $10,000. We desire to define the decision maker's risk-aversion utility function over that range. We use for this purpose the construction in Figure 6.5. On the horizontal axis of Figure 6.5 we have inscribed the range of possible outcomes, and on the right and left vertical axes, respectively, we have indicated measures of probability and utility. That is, we have arbitrarily assigned the point 0 on the utility scale to the utility of -$10,000 and the point 1 to the utility of $10,000. Imagine now that we desire to assign a point on the utility scale to the amount of $6,000. We proceed, in the manner of the previous discussion based on the axiom of continuity, to visualize a so-called reference lottery in which there are only two possible outcomes equal to the worst outcome of the possible cash flow being examined, or -$10,000, and the best possible outcome, $10,000.

-10

Figure 6.5

6 Utility, uncertainty, and theory of choice

125

We can deduce what would have to be the probability of success in the reference lottery, that is, the probability of receiving the $10,000 outcome, to make the expected value of the lottery equal to $6,000. This so-called breakeven probability follows from the equation p(10,000) + ( l - p ) (-10,000) = 6,000

(6.25)

The desired value of p is 0.80. Similarly, the probability of success in the reference lottery that would make the expected value of the lottery equal to, say, $4,000 is 0.70. If the probability of success in the reference lottery is 0.30, its expected value will be equal to —$4,000. In this way, we can build up a locus of "breakeven probabilities" that indicate the probabilities of success at which the expected value of the lottery will equal any designated amount within the range of possible outcomes of the cash flow, -$10,000 to $10,000. This locus of breakeven probabilities will coincide with the diagonal of Figure 6.5. We now suppose that the individual has received a certain cash flow of $6,000, and we ask him what would have to be the probability of success in the reference lottery to make him indifferent between retaining the $6,000 and surrendering it for a ticket in the lottery. He knows, of course, from the foregoing that the expected value of the lottery will be precisely equal to his $6,000 if the probability of success is 0.80. If, however, he exchanged his $6,000 for the lottery ticket, he would stand to gain a further $4,000 but would stand to lose $16,000, equal to the $6,000 he surrendered for the lottery ticket plus the $10,000 he would have to pay if the "lose" rather than the "win" result of the lottery occurred. Being a risk averter, as we shall suppose, the individual would state that he would require a probability of success greater than the breakeven probability to make him indifferent between his $6,000 and the lottery. Let us suppose that after a certain amount of questioning, meditation, and answering, the individual stated that the indifference would be established, and he would be prepared to exchange his $6,000 for the lottery ticket, if the probability of success were 0.90. In the same way, we might establish that the individual would be indifferent between the sum of $4,000 and the lottery, not if the probability of success were simply the breakeven probability of 0.70 but were, say, 0.85. If the individual had received zero dollars and he were invited to exchange it for the lottery ticket, or, that is, to accept a free ticket in the lottery, we can similarly ascertain the probability of success that would make the exchange possible. If, of course, the individual did accept the lottery ticket, he would stand to gain $10,000 but would also stand to lose $10,000. His risk aversion would, presumably, cause him to require a probability of success greater than the breakeven probability of 0.50. Let the required probability be 0.70. Finally, as an example on the other side of the range of possible outcomes of the cash

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II The neoclassical tradition

flow, we might imagine that an amount of -$4,000 had been received, or that a loss or a debt of $4,000 had been incurred. If the individual were to exchange that amount for a lottery ticket, the lottery organizer would, in exchange, be agreeing to relieve him of his debt of $4,000. If he were successful in the lottery, his net gain would be $14,000, equal to the sum of the $4,000 debt of which he had been relieved and the $10,000 prize in the lottery. But if the "loss" outcome of the lottery occurred, he would realize a net loss of $6,000, equal to the $10,000 he was now required to pay to the lottery less the $4,000 debt of which he had been relieved. His risk-aversion characteristics again will probably make him require a probability of success greater than the breakeven probability of 0.30. We can suppose it is 0.50. In Figure 6.5, these required probabilities of success in the reference lottery are shown above the relevant magnitudes on the horizontal axis. The same procedure may be followed with relation to any other magnitude in the cash flow range between -$10,000 and $10,000. If, then, these probabilities are joined as in the figure, we have what we shall call a locus of "indifference probabilities." It will be clear that the concavity of the indifference probability locus is due to the degree of risk aversion in the mind of the individual whose utility function is being derived. Moreover, the greater the degree of risk aversion, the greater will be the concavity in the indifference locus. If the individual were risk indifferent, he would have been satisfied at every point of the foregoing to exchange given amounts of cash for the lottery ticket so long as the breakeven probabilities were operative. The locus of indifference probabilities would then be linear and would coincide with the locus of breakeven probabilities. Finally, if the individual were a risk lover rather than a risk averter, his indifference probability locus would be strictly convex, or would swing down along the southeast of the breakeven probability locus in Figure 6.5. We are now ready for the final step that transforms the indifference probability locus into the desired utility function. Suppose, for example, that we wish to assign a utility number to the cash flow outcome of $6,000. We have just seen that the individual is indifferent between $6,000 and the lottery with a probability of success of 0.90. This indifference, then, requires us to assign to the amount of $6,000 the same utility number as represents the expected utility of the lottery with a probability of success of 0.90. That may be calculated as follows: E[U(L)] = 0.9(H/($10,000) + 0. \0U($ - 10,000) = 0.90(1)4-0.10(0) = 0.90

(6.26)

The utility number to be assigned to the amount of $6,000 is therefore 0.90. In a similar way, utility numbers may be derived for any amount within the

6 Utility, uncertainty, and theory of choice

127

range of cash flows between -$10,000 and $10,000. By assigning the arbitrary numbers of 0 and 1 to the limits of the range of possible cash flows, we have been able to read the indifference probability locus against the right vertical axis, or the probability axis, of Figure 6.5 and to read the same locus as the derived utility function against the left vertical axis, or the utility axis. This convenient result, of course, is due only to the arbitrary assignment of the utility range that we made. Any other numbers could have been assigned, as the utility function we have derived is, as we have indicated, an ordinal function. If the assumptions contained in the axioms we have defined are satisfied, and if individual behavior reflects the statement of the axioms or is in that sense rational behavior, the axioms permit us to conceive of a utility function of the general form shown in Equation (6.5). There, as in the instance we have just examined, the degree of concavity reflects the degree of risk aversion in the mind of the decision maker. Uniqueness of the utility function A significant difference exists between the ordinal utility function in terms of which we discussed consumer behavior earlier in this chapter and the utility function we have now derived. The former, we have seen, is unique up to a monotonically increasing transformation. But in the present analysis it is possible to envisage monotonically increasing transformations that, if they were applied to the investor's utility function, would not preserve a consistent ranking of portfolios. As a simple case, consider the following example, based on the information in Table 6.1. Column 2 of the table shows the utilities assigned to the range of values indicated in the first column, thereby describing an arbitrarily assigned ordinal utility function U\. In column 3 are shown the utility magnitudes that would apply if a monotonically increasing transformation of the first utility function were adopted, that is, U2 = (Ui)2. Let us suppose that two investment opportunities exist: Si, which offers a Table 6.1 Ordinal utility functions Dollars

Ux

U2 = (U{)2

0 2 4 6 8 10

10 40 65 85 100 110

100 1,600 4,225 7,225 10,000 12,100

128

II The neoclassical tradition

50 : 50 chance of receiving $2 and $10, and S2, which offers a 60 percent chance of receiving $4 and a 40 percent chance of receiving $8. From the second column of Table 6.1, describing U\, the expected utilities of S\ and S 2 are = 0.50(40)+ 0.50(110) = 75

(6.27)

E[U(S2)] = 0.60t/($4) + O.4 = 0.60(65)+ 0.40(100) = 79

(6.28)

It follows that 52 will be preferred to Si on the basis of its higher utility ranking. If the two investment opportunities are evaluated against utility function U2, the utility assignments are 6,850 for Si and 6,535 for S2. The order or ranking will now be reversed, and Si will be preferred to S2. Thus we see that a monotonically increasing transformation of the original utility function over the range of monetary values specified in the example may fail to be order preserving. It can be shown, however, that a monotonic linear transformation of the ordinal utility function will preserve the preference ordering in several important respects. Let us suppose a utility function of the form that provides, in accordance with Axiom 3, UB =PiUA + (1 ~Pi)Uc for some pt. Recalling that Y is a monotonic linear transformation of X if Y=a + bX, with ft>0, we can transform the utility function U to the function (6.29) It follows from this that U = (U*- a)lb, orU = cU* + d, where c = I/ft and d= —alb. We can say, therefore, that U B in the present example transforms into cU%-\-d. Following the example, we can therefore write +d

(6.30)

from which it follows that cU%=PicUX + c{\-pdUt

(6.31)

and therefore U$=p,UX + (l-pi)Ut (6.32) This demonstrates that the monotonic linear transformation of the original utility function given in Equation (6.29) provides the same results as the original utility function and is therefore itself a utility function.

6 Utility, uncertainty, and theory of choice

129

The utility functions provided by the von Neumann-Morgenstern analysis that we have followed up to this point, although they are in a basic sense ordinal, have to be regarded as cardinal in a restricted sense. They partake of the same limitations of ordinal measures that the consumer utility functions contain. They do not permit interpersonal comparisons of utility or permit the statement that an outcome providing a utility of, say, 70 is preferred twice as much as an outcome with a utility measure of 35. This follows, of course, because the choice of the origin of the utility scale is arbitrary. The von Neumann-Morgenstern utility numbers do, however, provide an interval scale such that the differences between the utility numbers are meaningful. This follows from the fact that a monotonic linear transformation preserves the relative magnitudes of differences between the numbers on the original utility scale. That is, if the difference Uc - UB is greater than UB — UA on an original scale, the same relative magnitudes of differences will be preserved under the monotonic linear transformation. Under a monotonic linear transformation, also, the sign of the rate of change of marginal utility, or the second derivative of the utility function, is invariant. The von Neumann-Morgenstern ''ordinal-cardinal" utility function, at the same time as it permits this comparison of utility differences, also permits the calculation of expected utilities in the sense in which we explored that earlier in this chapter. It thus leads to the decision criterion of expected utility, and it facilitates decision making under risk or the discrimination between stochastic objects of choice. The firm's choice of output under selling price uncertainty An application of the expected-value concept discussed in this chapter is found in the firm's choice of its optimum output when uncertainty attaches to the price at which that output can be sold.2 A brief digression on this problem will consolidate our understanding of many of the main points at issue in the following chapter. We assume that in a simple static model the firm is required to choose its output level x before the selling price p is known but on the understanding that the price will be determined in a perfectly competitive market. The firm's total cost function is assumed to be known as TC(JC)=A + C(JC)

(6.33)

where A represents the fixed costs and C(x) is the variable-cost function. The firm's decision maker possesses a utility function defined over profits, TT, where 7r=px-A-C(x) 2

(6.34)

The argument in the text is based on Sandmo (1971) as reported in Ford (1983, p. 173f.).

130

II The neoclassical tradition

As in the preceding sections of this chapter, the probability distribution of the selling price is assumed to be assigned by the decision maker and is denoted We can then make use of the expected utility therorem to specify the firm's expected utility of profit as E[U(7T)] = ^U[px-A-C(x)]f(p)dp

(6.35)

where p is assumed to be nonnegative, as it represents the market price of the output commodity. By differentiating Equation (6.35) with respect to x, we find the first-order condition for a maximum value of E(U). This is accomplished by first differentiating the utility term under the integral sign, applying the chain rule in doing so, to yield

F

{

\

dp

(6.36)

Equation (6.36) will be recognized as describing the expected value of the term under its integral sign to the left of the probability function [analogous to Equation (5.4) of Chapter 5]. We can therefore write, as the first-order maximization condition,

E[u'(7T){p-C'(x)}] = 0

C(X)>0

(6.37)

If, further, we differentiated Equation (6.36) with respect to x, we should have the second-order condition for the utility maximum: E[U"(TT){P-C'(X))2-

t/'(7r)C"(Jc)]D~ p dD It follows from Equation (8.2) that

158

II The neoclassical tradition

Dividing throughout by 1 — t yields

where X' equals the marginal productivity measure multiplied by a tax stepup factor to express it on a before-tax basis. The tax step-up factor, which also appears on the left-hand side of the equation, is the reciprocal of 1 minus the firm's tax rate. Both sides of the optimization condition in Equation (8.4) are expressing the firm's marginal cost of debt capital, and the left-hand side exhibits in more detail the "full marginal cost of relaxing the money capital availability constraint." The first two terms taken together describe the derivative of the firm's interest burden with respect to debt, d[r(K, D)]/dD, or the increase in the interest burden that results from the increased use of debt. We refer to this magnitude (r + D dr/dD) as the firm's marginal direct cost of debt. Recognizing the first term, r, as the interest rate that has to be paid on the marginal debt itself, the marginal direct cost of debt will be greater than the interest cost of the marginal debt. This is because the higher interest rate on debt capital that results from the increase in the debt-to-equity financing ratio will have to be paid, not only on the new debt now being issued, but also on the previously existing debt when that matures and has to be refinanced. Recalling the foregoing discussion of the equity owners' position, the minimum rate of return that must be earned on the marginal debt capital must exceed this marginal direct cost of debt by an amount sufficient to offset any tendency to a dilution of the market value of the equity. Consider, therefore, the following example. The magnitudes are hypothetical and no doubt exaggerated, but they exhibit the nature of the causation involved. Employing a time subscript of zero to refer to the firm's situation before the new financing and investment and subscript 1 to refer to the situation after the new financing, let us assume the following situation: D o = $3 million (the market value of debt capital) Vo — $6 million (the market value of equity) WO = DQ + VO = $9 million (the total market value of the firm's capital securities) ro-0.05 DI0 = $150,000 (debt interest) p 0 = 0.10 Number of equity shares outstanding = 100,000 Earnings per share (EPSo) = $6 Market value per share (So) = $6/0.10 = $60

8 Cost of money capital: further analysis

159

Table 8.1 Income statement data ($000)

Operating income Less debt interest Earnings before tax Less tax

Before new financing

Required position after new financing

Incremental data

$1,350 150

$1,800 300

$450 150

1,200 600

1,500 750

300 150

Net income (Residual equity earnings)

600

750

Earnings per share

$6

$7.5

Market value per share

$6/0.10 = $60

$7.5/0.125 = $60

150 $1.5

Marginal debt financing (AD) — $2 million ri = 0.06 DIi = $300,000 (after refinancing all maturing debt) P!= 0.125 ^ = ^=0.50 In this situation, the full marginal cost of debt as defined on the left-hand side of Equation (8.4) quantifies as ^

^

J

3

^

r

0.06 + 0.015 + 0.15 = 0.225

(8.5)

The full marginal cost of debt, or the required rate of return on a before-tax basis necessary to establish economic viability, is 22.5 percent. To reinforce the understanding of these relationships, consider the pro forma income statement of the firm in Table 8.1. The data in Table 8.1 imply that the residual earnings per share of equity after tax would increase by $1.50, sufficient to increase earnings per share to $7.50. When this amount is capitalized at the new higher capitalization rate of 12.5 percent, the market value per share of $60 can be maintained, and no equity dilution will have occurred. But this will occur only if the operating income of the firm increased by $450,000, or by 22.5 percent of the new $2

160 II The neoclassical tradition million investment, precisely the gross before-tax required rate of return or the full marginal cost of the debt as specified in Equation (8.5). Weighted average cost of capital Leaving aside now the question of computation, and ignoring the technical adjustment for the firm's tax rate, we recall the firm's full marginal cost of equity that was brought into relation with the full marginal cost of debt in Equation (4.47). At the firm's optimum structural planning point, both these full marginal costs were equal to the magnitude p \ , which therefore emerged as the firm's optimum planning discount factor or valuation rate. We consider now whether any traceable relation exists between this planning discount factor and the more familiar concept of the firm's weighted average cost of capital. It will be shown that the two are in fact equal to one another when the firm has been brought to its optimum size and structure. At that point, and when all parts of the interdependent planning nexus have been brought to their optimum solution values, the weighted average cost of capital will be equal to both the full marginal cost of debt and the full marginal cost of equity. It can therefore be employed for valuation purposes at the margin. This will, in fact, provide a potential linkage with the results of the general equilibrium-theoretic analysis. But it follows from the full set of optimum conditions that this is admissible only when the firm has chosen its mutually determined optimum levels of output, factor usages, asset investments, and financing sources. The reasons for this will be clarified in the following argument. Consider the following notation: V, IT, and p refer, as before, to the market value, profits, and capitalization rate applicable to the owners' equity, and D, F, and r refer to the market value, interest earnings, and market yield (average rate of interest) on the debt capital of the firm. In the planning model of the firm with which we are working, the market value and the book value of the debt will be equal. This is because we are supposing that following any change in the amount of debt, or in the average rate of interest on the debt, all of the firm's previously existing debt, as in the foregoing numerical illustration, is refinanced at the new rate of interest. Finally, W and O refer to the total capitalized value, W, of the net operating income of the firm, O, and the relationship O/W will provide a measure of /, the effective rate of capitalization at which the total earnings of the firm are being valued in W. The total value of W, moreover, will be equal to the sum of D and V. These definitions imply the following relations. V =7T/p

7T=pV

(8.6)

D=F/r

F=rD

(8.7)

8 Cost of money capital: further analysis

W = O/i

O=iW

161

(8.8)

If, at any planning point, a set of security values and income statement items are observed and described as in the foregoing list, the following definitional statements can be made. l

(89)

-w~~ttT

whence

From Equation (8.10), the overall capitalization rate, /, which may now be referred to as the average cost of capital, is seen to be a weighted average of the costs of debt and equity capital, r and p, respectively. Each such capital cost is weighted by the relative importance of the corresponding capital in the total market value of the firm's securities. In the optimization model, the debt and equity capital costs were assumed to be functionally related to the amounts of debt and equity capital measured at book values (as book values represent the decision variables confronting the firm1). Equation (8.10) may therefore be written as i = r(K,D)^

+ p(K,D)^

(8.11)

We are interested now in the combination of debt and equity capital at work in the firm, D and K, that will afford the minimum attainable value of / when the firm is at its optimum capital usage point. We accordingly differentiate Equation (8.11) partially with respect to D and K, setting the partial derivatives equal to zero, and noting that W-D + V. We note also that because we are investigating these relationships at the optimum planning point, we are at the situation at which, as seen in Equation (4.51), dVldK is equal to unity. Employing this fact, dW/dK may be replaced in what follows by BW/dV. Differentiation of Equation (8.11) yields

1

An additional reason for using book values rather than market values as functional arguments is that the use of market values introduces a bias due to the fact that they reflect business risk (on which we shall comment further later) as well as financial risk. This implies that market values may vary, quite independently of financial risks, in reaction to changes in business risk, as the latter induces changes in the debt and equity required rates of return or capitalization rates. See Barges (1963), Turnovsky (1970, p. 1065), and Herendeen (1975, p. 125).

162

II The neoclassical tradition

Ddr

di

D , Vdp

Multiplying Equation (8.12) by W and rearranging yields

r+D

w+v^=4+pw

(814)

Similarly, Equation (8.13) yields

Equations (8.14) and (8.15) together imply that when full optimization conditions are satisfied, the full marginal cost of borrowing and the full marginal cost of equity are both equal to the weighted average cost of capital. It is instructive to compare the results in Equations (8.14) and (8.15) with Equations (4.47) and (4.48). Combining the equations, it follows that r RM]

(8.34)

and

yB

(8.35)

Substituting these equivalences into Equations (8.32) and (8.33) provides E(RA) = RF + ^- Cov(X A , RM)

(8.36)

y

(8.37)

and ^wy V * A , RM)

It is known from previous development that the rate of return on the levered equity shares of firm B can be expected to exceed the rate of return on the unlevered equity shares of firm A. We therefore subtract E(RA) from E(RB), or Equation (8.36) from (8.37), to yield

174

II The neoclassical tradition E(RB)-E(RA) = \x Cov(XA, RM) - ± I ^

(8.38)

If, now, Equations (8.32) and (8.33) are solved for E(XA) and the results equated, it follows that VA[RF + n Cov(RA, RM)] = VB]^i

COV(RB,

RM)+RF

y

j (8.39)

In this expression, we can substitute the equivalences in Equations (8.34) and (8.35) to provide +^

COV(XA,flM)]= VB[^ COV(XA, RM)+RF

M ^ j

(8.40)

From Equation (8.40), it follows that (8.41) which implies that VA = V B + D B

(8.42)

The significant result in Equation (8.42) states a principal conclusion of the asset pricing model from which it has been derived. It is that when the full equilibrium conditions are satisfied, the total market value of the levered firm B, shown as the sum of the market values of its equity and debt capital on the right-hand side of Equation (8.42), will be equal to the market value of a comparable unlevered firm, as shown on the left-hand side of the equation. This is the same result as was established previously under the assumptions of a smoothly functioning arbitrage mechanism in a perfect financial market. The total market values of comparable levered and unlevered firms, according to this vision, must be equal and invariant with respect to their financing mix. Making use of this result, Equation (8.38) implies that E(RB)-E{RA) = fi Cov(XA, RM) ^ y

(8.43)

Inspection of Equation (8.36) indicates that in the generalized equilibrium the asset market model has in view, fi Cov(XA, RM) = [E(RA)-RF]VA

(8.44)

and Equation (8.44) may then be substituted into Equation (8.43) to yield -RF] ^

(8.45)

8 Cost of money capital: further analysis

175

Rearranged, Equation (8.45) provides E(RB) = E(RA) + [E(RA)-RF] ^

(8.46)

The conclusion in Equation (8.46) states that in full general equilibrium conditions the rate of return on the levered firm's equity will be equal to that on the unlevered firm's equity plus a risk premium. This risk premium is equal to the market's risk premium on the unlevered firm's equity [E(RA)—RF] multiplied by the financial leverage ratio in the levered firm. This statement can be compared with the previous conclusion derived from the ModiglianiMiller entity theory of capital costs and stated in Equation (8.23). It is reproduced for convenience of comparison: p = i + (i-r)DIV

(8.47)

By comparing Equations (8.46) and (8.47), it is seen that both bodies of analysis lead to the same conclusion. The equilibrium-theoretic analysis of a perfect money capital market provides the same result for the cost of equity capital as the entity theory of money capital costs under the assumption in the latter theory of a freely functioning market arbitrage mechanism. Growth of the firm and the cost of equity capital The value of a share of common stock has been defined up to this point as the present discounted value, or the present capitalized value, of the future expected earnings accruing to the owner of the stock. In our optimization model, the value of the equity has been stated as V=ir/p, where TT refers to the residual income accruing to the owners and p describes the owner's required rate of return or capitalization rate. This analysis, however, has been cast for the main part in a static mold. The different approaches to the money capital and valuation problem that we have discussed have also been developed from the same static perspective. We did observe, in the introductory discussion of the firm's economic position and performance statements in Chapter 2, that not all of the residual earnings of the firm may actually be paid to the owners in the form of dividends. Part may be retained and reinvested in the firm, and the market value of the ownership stock may increase as a result, as the future prospective income stream rises and reflects the earnings on the reinvested capital. The valuation formula we have used may be accommodated to the possibility of the growth of the firm that results from such reinvestment. Let us first restate the economic value of a share of common stock as the present discounted value of the future stream of cash flow benefits that the owner of the stock expects to receive. The value of a share of stock may be defined as

176

II The neoclassical tradition ^

^

-

-

(8 48)

-

-

where the series of terms in the summation extends to infinity. Each term in the series in Equation (8.48) expresses the present discounted value of a dividend payment, where Dt describes the dividend to be received in year t. The value of the share of stock can be stated alternatively as

lf



1 1

/

Figure 11.1 larger the level of contemplated investment expenditure, the lower will be the expected rate of return on the marginal, or the last added, project. Consider now the supply-of-funds curve, or the marginal-cost-of-moneycapital curve, MS in Figure 11.1. Note that a kink appears at point A. The portion of the curve extending from the vertical axis to point A describes the supply of money capital funds from internally generated sources. That is shown in the figure as a positive function of the effective cost of funds, r. The AS portion of the supply curve describes the rate at which, and the money capital costs at which, investable funds are available from the external capital market. The cost of internally generated funds or retained earnings is shown in the figure to be lower than that of externally available capital. In the discussion of the investment project decision in Chapter 9, it was observed that the cost of retained earnings will approximate that of new equity issues, or the rate of return required by the equity holders. This followed from the fact that the retention of earnings is a substitute for new equity issues, as both are sources of equity capital. But the equity holders might enjoy a tax advantage from earnings retention compared with the distribution of dividends. The retention of earnings, provided the funds retained are invested effectively by the firm, can be expected to lead to an increase in the market value of the equity shares. An investor could realize part of that increased value by selling off part of his holding, thereby making himself subject to capital gains tax on the realized capital appreciation. That gains tax, however, will in general be lower than

11 Production and the place of money capital 199 the personal income tax that would have been paid on the receipt of dividends. This relative tax advantage of retained earnings might therefore reduce their effective cost marginally below the cost of new equity funds. Moreover, new equity issues carry further costs that are avoided in the use of retained earnings. A new issue of equity will frequently require the new shares to be sold at a fractionally lower price than the current market value of the existing shares, such an underpricing being necessary to guarantee the successful flotation of the issue. Also, a certain amount of underwriting costs will be incurred. No such costs exist in the case of retained earnings, and this, together with the tax advantage, suggests a lower effective cost of internally generated funds. The retained earnings portion of the supply-of-funds curve in Figure 11.1 has been shown as rising slightly. This reflects the possibility that increased earnings retention, implying a lower dividend distribution, may cause investors to fear that the risks in the firm's operation may be adversely affected as investment proceeds further. If that is so, the required rate of return on the equity will increase, leading to the possibility of a dilution of the value of the equity holders' investment position. We examined this kind of effect when we observed the possible impacts from the marginal use of debt capital. There is no a priori necessity for these incremental risk effects to occur in the earnings retention case. But the incremental income streams that the retained earnings will generate remain in the future; and the further in the future they are, the more risky they may appear to be. The distribution of earnings as dividends, on the other hand, provides the equity holders with immediate cash benefits. But any such risk effect depends on the quality of the investment opportunities in which the management invests its funds. It has been argued by exponents of the equilibrium asset market theory that in perfect capital markets investors will be indifferent between dividend distributions and retained earnings (see Brealey and Myers, 1984, p. 336f.). This so-called dividend irrelevance theorem is based on the assumption that a firm's capital expenditure program has been decided upon and that earnings retention and new capital issues are then two alternative ways of financing the given capital budget. Quite apart from whatever difficulties attach to the assumptions of the perfect capital market theory, investors who might be concerned about the incremental risks associated with the investment of retained earnings may visualize a somewhat different set of relations. Rather than contemplating the use of retained earnings and new equity issues as simply alternative ways of financing a given capital budget, they may be concerned also with the way in which earnings retention may lead to investment decisions by the firm's management that would not otherwise have been taken. The MA portion of the supply-of-funds curve in Figure 11.1 may also reflect an increase in the cost of internally generated funds that may occur if, as

200

III Postclassical perspectives

we shall examine below, the firm's selling price is raised in order to increase the availability of funds from residual cash flow sources (see Eichner, 1985, Ch. 3). This points to an examination of the relation between the firm's average cost markup, which translates those average costs into the firm's selling price, and the effective cost of changing that markup if investment or financial policies should make that desirable. The estimation of the marginal cost of external money capital, or the costs that are described in the AS portion of the supply curve in Figure 11.1, may proceed in the manner we indicated previously. Rather than relying on the assumptions of the equilibrium asset market theory, we may look directly to the marginal costs of debt and equity funds, or a combination of them, under imperfect capital market conditions. In other words, we rely again on what we exhibited earlier as the full marginal cost of relaxing the firm's money capital availability constraint. This may or may not, we have seen, be precisely measurable by the firm's weighted average cost of capital. If there is reason to believe that the firm is proceeding along an equilibrium growth path, and if, therefore, the firm raises its incremental finance in such a way as not to cause any change in its financing mix, and if it is maintaining production structures that provide the same degree of business risk, the weighted average cost of capital will be relevant and usable. But if, on the other hand, investment is being undertaken as a means of changing the firm's growth rate, perhaps by diversifying into other sectors of the economy where higher growth rates are attainable, a careful calculation will need to be made of the new overall risks that will be involved and the new implied costs of money capital. To the extent that firms do make new capital issues, the total supply of securities available in the financial asset market will be increased. The significance of this lies in the fact that in the exposition of the equilibrium asset market theory of Chapter 7 the assumption was made that the total supply of securities did not change. If firms do make new capital issues, the specification of the investors' asset opportunity set in the equilibrium asset market theory will be changed, and a different optimum market portfolio will be defined as a result. This will change the locus of the efficient boundary of the asset opportunity set, and it will alter, therefore, the characteristics of the Capital Market Line. This, in turn, will cause a change in the equilibrium market price of risk and a change in the effective costs of capital to all firms, or in the required rate of return on each firm's risky capital securities. In the situation envisaged in Figure 11.1, the firm would undertake an investment expenditure of 0/*, of which ON would be financed from internally generated cash flows and the balance, NI*9 would be financed by new capital security issues. Our task now is to develop a model of the firm in

201

11 Production and the place of money capital

MC

•AR

ATC = AVC + AFC AVC

60

80

100

% ERC

Figure 11.2 which the underlying determinants of these relations can be brought into clearer focus.1 Operating characteristics of the firm Let us imagine a firm whose production decisions are made on the basis of a fixed technological coefficient production relation. It will combine factor inputs in fixed and, in the short run, well-defined and unalterable proportions. We now depart from the earlier neoclassical assumptions of infinitely divisible and continuously substitutable factors of production. We may imagine that the firm is, in the expressive language of Eichner (1976), a "megacorp." It will conceivably arrange its production apparatus in such a way as to install a number of plant segments, each of which will operate with fixed factor combinations and will exhibit, as a result, a cost level dependent on its optimally attainable productivity. That will be determined by the technological character and efficiency of the particular vintage of capital equipment it contains. The operating characteristics of the firm can then be depicted in Figure 11.2. The horizontal axis of Figure 11.2 describes the Engineer Rated Capacity 1

The following sections are heavily indebted to Eichner (1976, 1985), whose work has profoundly significant implications, not only for the theory of the firm and its financing, but also for problems in macrodynamic theory and policy. Figures 11.2, 11.3, and 11.4 and the analysis to which they are related are essentially an interpretation of Eichner's work, and full acknowledgment is due to his important conceptual and terminological inventions. See also the related and highly insightful discussion in Wood (1975).

202

III Postclassical perspectives

(ERC) of the firm's production apparatus. For production levels below 100 percent of ERC, the average variable cost per unit of output remains constant. This follows from the fact that the firm, as suggested, will be likely to maintain a number of plant segments, each of which will be brought into operation or taken out of operation as the firm desires to alter its total level of production. Each such production segment, however, will have been designed to operate at maximum efficiency, or at a lowest attainable average cost level, depending on its given vintage and technological characteristics. If it is called into operation, it will normally be operated at that optimum level of intensity. In this manner, the firm will be able to maintain a fairly constant level of overall average variable costs per unit of output. Or, perhaps, the average variable-cost curve depicted in Figure 11.2 might rise as the firm nears its 100 percent ERC, as older and less efficient plant segments are brought into operation in order to achieve the desired level of output. In the general case, a firm may aim to operate at a usual intensity of, say, 80 percent of its ERC as shown in the figure. Plant design may be such as to incorporate a target standard excess capacity. This, in turn, will provide the firm with the degree of production flexibility it may imagine it needs in order to satisfy the requirements of the output fluctuations it expects to encounter during normal business fluctuations. The average fixed cost per unit of output will, of course, decline as the level of output increases. When this is added to the average variable cost, the average total cost curve, ATC, is defined as in Figure 11.2. The marginal cost, which will again be constant (and equal to the average variable cost) at output levels below 100 percent of ERC for the reasons previously adduced, will rise if output is pushed beyond the 100 percent ERC level. The level at which the firm plans to operate can be referred to as its Standard Operating Ratio (SOR). It will be possible for the firm to determine at its planning stage the average fixed cost, average variable cost, and average total cost at the SOR level. The firm then faces the decision as to the price at which it should offer its output for sale. At that point, it must bear in mind that if it is going to maintain its share of its output market, it must plan to grow at the expected growth rate of the industry of which it is a member. This may be determined by estimates of the growth rate for the economy and by the likelihood that new entrants to the industry might succeed in reducing the firm's attainable market share. If, however, the firm is to retain its market share, and if it is to maintain the established degree of real capital intensity in its production arrangements, it will need to expand its capital stock each period at a rate equivalent to its expected overall rate of growth. This, then, will provide the firm with an estimate of its desired or target rate of incremental investment per period. Of course, other reasons may exist for incremental capital investment in

11 Production and the place of money capital

203

the firm. Investment may be made for purposes of achieving cost reductions, to install technologically more efficient or newer vintage capital equipment, or to diversify into new sectors of industry. The latter possibility introduces some potentially important analytical considerations. First, the firm may desire to diversify into a more rapidly expanding industry in order to increase its overall rate of growth. Or it may wish to diversify as a means of maintaining its growth rate if, for any reason, the attainable rates of growth in its currently established industry have slackened. But second, the suggestion that diversification might be desirable contradicts a significant conclusion of the equilibrium asset market theory. That theory implies that firms cannot accomplish anything positive for investors by undertaking diversification and that, indeed, the best interests of investors are served by leaving them to accomplish their own diversification by combining portfolio assets in the manner we have examined. In any event, the firm will decide, given the cost relations exhibited in Figure 11.2, the size of the markup it will add to its average total cost, as estimated at its SOR production level, to determine its selling price. The price it sets and announces to the market will then be defined as its average total cost multiplied by a markup percentage. We define the markup percentage as m and derive the relation between average total cost C and the selling price P: m = (P - QIC,

(11.1)

P = (1 + m)C

(11.2)

whence The apparent simplicity of this relation should not be allowed to conceal its significance. When production volumes are maintained at the SOR level on which the cost estimates and the selling price decisions were based, the firm will be realizing its target rate of return on the total money capital it employs. An alternative way of referring to its desired markup factor, then, is to say that it will mark up its average costs, and thereby set its selling price, in such a way as to realize a target rate of return on capital at its standard operating volume. The basic reason it does this, it should be borne in mind, is that it can thereby expect to realize also the internally generated cash flow it plans to use for financing its annual investment budget. If, in the course of business fluctuations, the demand for its product should increase, the firm's production can be increased above its SOR level without any change in its selling price. This, of course, is quite contrary to the vision of market price and volume changes implicit in the neoclassical view of industrial operations. In the present case, we have the prospect of price rigidity and volume flexibility. But at the same time, if the output volume is increased

204 III Postclassical perspectives in the manner envisaged, there will be an increase in the residual internally generated cash flow, or in the ACF magnitude (Average Surplus Cash Flow) in Figure 11.2. In that case, a larger amount of funds will be available for investment or for distribution to the equity holders in the form of dividends. Alternatively, depending on the level of incremental investment envisaged, the higher internally generated cash flows may diminish the need for reliance on new issues of securities in the external capital market. Moreover, if the production volume rises above the SOR level, the realized rate of return on the money capital invested in the firm will also rise above its previous target level. By the contrary argument, of course, if demand should decline and production levels fall below the SOR level, both the internally generated cash flow and the realized rate of return on capital will fall. In that event, the firm will have to decide whether it should reduce its capital budget or increase its demand on the external money capital market. Or it may, depending on whether it thought its diminished cash flow was a temporary phenomenon, increase its demands on the financial institutions for short-term loans, with a view to repaying them when the hoped for revival in demand and cash flows was realized. The extent to which that can occur, however, will depend on the creditworthiness of the firm in the short-term loan market, on the willingness of the financial institutions to supply the short-term accommodation, and possibly on the conduct of monetary policy that sets the tone and the overall availability of funds in the loan market. The firm may not, however, enjoy complete freedom to set its selling price in the manner we have suggested by marking up its average cost to realize a target rate of return on capital. It may be a price follower in its industry, not a price leader. Its demand curve, or its unit selling price curve, shown as AR in Figure 11.2, may be set at its existing level by the industry's price leader. In that event, the firm will decide on the standard operating volume at which it should plan to produce in order to achieve its target rate of return and cash flow. It follows that the average rate at which surplus internally generated cash flows become available differs from the marginal rate. The former is, as indicated, the difference between the selling price and the average total cost. But the marginal surplus cash flow is the difference between the selling price and the average variable cost. In conditions of rising demand, the average surplus cash flow will increase proportionately more rapidly than the marginal cash flow, with a contrary relation existing under conditions of reduced demand. We note finally the implications of this view of the firm's operations for the degree of risk to which it is exposed. If the firm is able to maintain its established share of the market or markets in which it sells its output, its principal risk will be that associated with demand fluctuations in the industry

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market as a whole. Of course, there remains the risk that in the context of the growth of the economy and the industry, the firm may not be able to maintain the market share. The risk that new entrants to the industry may capture part of the firm's existing market share will be one of the factors that the firm takes into account in deciding whether or not it should make changes in the average cost markup by which its selling price is set. Moreover, there are also risks associated with fluctuations in the economy. These give rise to the fluctuations in demand that we considered above when examining their possible effects on the realized cash flows and rate of return on capital. In the event of such fluctuations, the degree of instability in the residual income stream earned for the owners will be determined by the overall degrees of what we referred to in Chapter 2 as operational leverage and financial leverage. Money capital employment decision The foregoing analysis implies that the incremental capital investment decision in the firm is integrated with its production and pricing decisions. The selling price at which the firm offers its output in the market will be set at that markup over cost that provides it with the internally generated cash flow it needs, along with the external financing it is prepared to undertake, to finance its capital expenditure. Let us suppose that in an established and ongoing situation the firm is generating precisely the amount of internal surplus cash flow it needs to implement its regular capital expenditure program. This, we may suppose, enables it to maintain its normally desired rate of growth, consistent with the expansion of its industry and with its established market share. We may consider, then, the possible ways in which the firm might finance a new set of investment opportunities in order to undertake a faster than previously planned expansion. How, in other words, should the incremental money capital be divided between larger internally generated cash flows and demands on the external capital market? Given the argument summarized in Figure 11.2, a higher average surplus cash flow could be generated by increasing the markup factor, say from m as in Equation (11.2) to m + Am. If the firm were able to operate at its previously established level of output, or at the same SOR level, the higher selling price implicit in the higher markup factor would, of course, generate higher internal cash flows. But whether the firm could maintain that production level will depend on the elasticity of demand for its output. It will actually depend on a number of such factors that we might inspect in the following way. Consider for this purpose Figure 11.3. In Figure 11.3(a) we have shown a relation between a possible increase in the markup factor m on the horizontal axis and the expected incremental surplus cash flow per period of time, Fit, that might result from such a decision.

206

III Postclassical perspectives AF/t

(AF/t),

Am

Figure 1 1 . 3

It is posited in the figure that increases in the markup factor will be likely to increase the surplus internal cash flow at a decreasing rate. There are a number of reasons why this might be so. First, the firm will have to consider, in its estimate of the outcome following a change in the markup factor, the extent to which the higher selling price will affect its position in the output market. It must take account, that is, of the substitution effects that might divert demand onto other competitive products, depending on the crosselasticities of demand that exist, as well as on the direct price elasticity of demand for the products of the industry of which the firm is a member. If the industry demand curve is inelastic, an increase in a selling price that is followed by the industry in general will increase the total revenue from product sales. A price increase under conditions of elastic demand, on the other hand,

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will lead to a reduction of total sales revenues. The firm that is contemplating a possible increase in its markup factor will therefore make some estimates of the likely impact of these elasticity effects, particularly as they stem from commodity cross-elasticities of demand. Second, the firm will need to consider the likelihood that if it increases its markup and selling price, and if it increases its revenue by continuing to produce at its existing SOR production level, it will generate a larger rate of return on the capital employed in the firm. This higher rate of return might not be able to be realized permanently, of course, because the longer-run effects of its actions might offset any shorter-run benefits. The industry of which the firm is a member may be confronted in its immediate market by an inelastic demand curve, and firms in the industry may therefore be able to increase their revenues by increasing their markup factors. But in the longer run, the cross-elasticities of demand for substitute commodities could operate in such a way as to lower the industry's demand curve and thereby diminish every firm's revenue-generating prospects. But whereas attention is thus paid to the longer-run as well as the shorter-run effects, the prospect of a higher rate of return on capital may nevertheless attract new entrants to the industry. If the firm has reason to believe that such new entrants may emerge and in due course reduce its existing market share, that will also be taken into account in deciding on the wisdom of increasing the markup factor and the selling price. The extent to which such new entry problems may exist will depend, among other things, on the minimum size of the initial capital investment necessary to obtain a foothold in the industry as well as on other entry barriers such as access to input materials and complementary production resources. We may suppose, therefore, that the firm, in facing the decision whether it should increase its markup factor and selling price with a view to increasing its internal surplus cash flow for investment purposes, considers the following possible developments over time. It may estimate that in the short run, say the next few years, it may be able to increase its surplus internally generated cash flow by a certain amount if the markup is increased. But it estimates also that after that time it will begin to experience a reduced cash flow because of the kinds of considerations we have referred to. On the basis of projected data of this kind, it can perform a cost-benefit analysis, or an economic valuation or rate-of-return analysis of the kind we considered in Chapter 3 and other places. The anticipated ultimate reduction in the cash flow below what it would otherwise be will assume a calculable time shape. Similarly, the nearer term increase in cash flows generated in the firm will assume an estimated time shape. It is then a straightforward matter to compare the size and time shape of the ultimate cash flow reductions with the size and time shape of the im-

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AF/t

Figure 11.4

mediate cash flow benefits. The former can be regarded as the cost of realizing the latter. It is possible, therefore, to set these two estimated incremental cash flow streams, one positive and the other negative, against each other and to calculate the rate of discount that will reduce them both to the same present discounted value. The discount factor that establishes the equality will be an estimate of the effective cost of realizing the benefits defined in the relation between the two cash flow streams. The vertical axis of Figure 11.3(a) shows the annual equivalent incremental cash inflow that, it is anticipated, would result from different possible increases in the markup factor. In Figure 11.3(b), the vertical axis depicts the effective cost of realizing those incremental cash flows by setting against the increase in the markup factor that would give rise to them the effective cost, or the effective rate of discount as described in the preceding paragraph. Taking Figures 11.3(a) and (b) together, the firm now has, on the vertical axes, estimates of both the incremental annual cash flow and the effective rate of cost associated with any specified change in the markup factor. We may then refer to the effective rate of cost as the cost of internally generated money capital. The data on the vertical axes of these figures are brought together in Figure 11.4. Take first the curve OAC in Figure 11.4. This describes an effective supply curve of internally generated money capital funds, understanding that this analysis is intended to apply to incremental possible cash flows additional to the regular ongoing capital budget expenditures that are financed from exist-

11 Production and the place of money capital 209 ing markup and pricing relations and the established use of external capital market sources. The curve can be expected to assume the convexity illustrated in Figure 11.4 by virtue of its derivation from Figures 11.3(a) and (b), where the concavity of the incremental funds curve in Figure 11.3 (a) is married to the convexity of the effective-rate-of-cost curve in Figure 11.3(b). Inscribed in Figure 11.4 also is an incremental supply of money capital curve, HAS, describing the costs at which further amounts of capital may be obtained from the external capital market. This has been drawn with a positive slope to suggest that the capital market may impose larger required rates of return or higher effective costs of money capital on the firm as its capital expenditure continues to rise. Questions of risk that we have considered at length already come into view. If, as the level of capital financing increases, the firm makes increasing use of debt capital funds, the implications of the increased financial leverage can be expected to increase the cost of equity capital as well as raise the cost of debt, imparting an upward tilt to the cost of external capital funds curve. Much would depend also on the nature of the risks attaching to the firm's proposed incremental investments if it is investing as a means of gaining entry to a new industry, in which, conceivably, it does not have a history of experience and management. Moreover, the same effects as previously will again follow from the operational leverage and financial leverage that enter the firm's production and financing structures as a result of the incremental investment and financing. We introduce to Figure 11.4 also the firm's demand curve for incremental investable funds, understanding again that we are here describing the demand for funds over and above those provided for by the established markup and pricing policies and capital market relations. This demand-for-funds curve is juxtaposed with the effective supply-of-funds curve. The latter will be described by the internally generated cash flow curve extending from the origin to the point A, and thereafter by the AS portion of the external funds supply curve. The dotted portion of the internal funds supply curve, AC, is irrelevant because the cost of funds from that source has increased, beyond point A, above the external cost. Similarly, internally generated funds will be employed up to point A, as the cost of funds from internal sources is lower in that range than that of externally available money capital. The effective-supply-of-funds curve is therefore described by OAS in Figure 11.4. The optimum level of incremental investment, reading from the supply-and-demand cross at point B, will be (W. Of this amount, OAf will be obtained from internal sources, with the increase in the markup factor being set at the level that makes that possible, as derived from Figure 11.3(a). The balance of incremental funds, MN in Figure 11.4, will be obtained from the external capital market at an effective marginal cost of NB. Of course, the incremental demand-for-funds curve, D, in Figure 11.4 could

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have crossed the effective supply curve OAS in the segment 0A. All of the necessary funds would then be obtained by making the change called for in the firm's markup factor and selling price, and no additional recourse would be made to the external capital market (see Eichner, 1976, 1985). It is conceivable that in the presence of temporarily surplus internally generated cash flows the firm could be a supplier of funds to the capital market. The firm's optimum-structure decisions The analysis we have developed in this chapter can usefully be compared with that in Chapter 4. In that earlier chapter we explored the ways in which, after refurbishing and expanding the widely accepted neoclassical theory, it was possible to conceive of a simultaneous solution to the firm's production, pricing, investment, and financing problems. The principal result of that analysis was the realization that the firm's optimum production structure was not independent of its optimum financing structure and that this carried along with it a fundamental message regarding the specification of the firm's effective cost of money capital. That was defined as the full marginal cost of relaxing the firm's money capital availability constraint. This latter concept can be carried over to the level of analysis contained in this chapter, as can also the basic notions of the valuation of income streams and the implicit rates of return on capital investment outlays. But some vital differences exist. In the earlier neoclassical analysis, no consideration was given to the possibility of earnings retention in the firm and the division of money capital sources between internally generated funds and financing obtained from the external capital market. That question was submerged in the light of the fact that the model of Chapter 4 was designed as a long-run planning model, which could, however, be reconsidered at any desired intervals of time. No division of the total equity funds into internal and external sources was made because, as we have seen, the costs of those funds are comparable, being close substitutes for one another. In the argument of this chapter, on the other hand, that division of fund sources has been brought into critical prominence. This has been possible for a further significant reason. In this chapter the firm has been regarded not so much as a static entity that structures itself once and for all at a point in time but as a developing structure that needs, if it is to maintain its place in a growing economy, to be continually expanding. It needs to reassess in each time period its optimum rates of capital investment and money capital usage, along with its most advisable financing sources, as well as reconsider its pricing policies, via its average cost markup policies, that make the realization of its overall goals possible. But one final point of similarity between the methodologies of Chapter 4 and the present chapter remains. In neither of these analyses is it intended to

11 Production and the place of money capital 211 leave the impression that the functional relations and the marginal values of outcomes, such as the comparison between the marginal efficiency of investment and the marginal effective cost of money capital in this chapter, can be specified with uniquely definable accuracy. In all of this work on the questions of the optimum structures and the decision problems of the firm, the realities of uncertainty and the relevance of the ignorance and the unknowledge that abounds impinge in ways we have not yet investigated directly on the making of management decisions. Our arguments have been designed to alert the analyst to the main directions of causation in some of the more critical relations that exist, or must be seen to exist, in the theory of the firm and its employment of money capital. In the following chapter we shall turn explicitly to the construction of a framework for decision making that addresses the uncertainties with which real historic time is inevitably pregnant.

CHAPTER 12

Uncertainty and decisions in the firm

The analytical issues of time and uncertainty have been relevant to many parts of our exposition. They throw an important light on many aspects of sequential decision making in the firm. In the traditional development of our subject, the assumptions that have been made in order to render the problems of time and uncertainty tractable have been related to the notion of equilibrium and equilibrium theorizing. In this chapter we shall reexamine a number of the interrelations that exist between these issues, and we shall bring into focus again a number of the questions that were raised in this connection in Chapter 1. The theory of the firm has all too often been cast in a timeless, static, certainty, or certainty-equivalent mold. The capitulation of economics to the analytical priorities of general equilibrium theory has left the theory of the firm preoccupied with the description of equilibrium states of affairs. This has become paramount so far as our principal object in this book is concerned, namely the analysis of the firm's employment of money capital. Money capital costs have been conceptualized by the theory as definable by the rates of return that investors require on risky assets when equilibrium conditions are satisfied in the financial asset markets. We have suggested, on the other hand, ways of envisioning the money capital problem from the perspective of partial equilibrium and imperfect market theory. Our move in that direction comes from the recognition that the general equilibrium theory, under its perfectly competitive assumptions, has no way of explaining how decisions are made in nonequilibrium situations. That, of course, is the only kind of situation in which decisions have to be made. If equilibrium obtains, the sea is calm and the ocean is flat and all is at rest. If equilibrium does not exist, the theory has no way of telling us how we can get into equilibrium. We know it only if we are there. The realities of time, moreover, bound us in a relative ignorance and shatter the comfortable epistemological security that assumptions of certainty and certainty-equivalents provide. Knowledge cannot be known before its time. But reality forces upon us the unavoidable moments of economic decision, and in them we form imaginative perceptions of the possibilities that the future contains. We inherit at our decision moments an accumulated knowledge and awareness and all the epistemological baggage that uniquely identifies us. 212

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We bring to those moments our particular complex of resources, endowments, skills, and capacities. We inherit a knowledge, or at least our private interpretation, of the fact situations that have structured the course of history that antedates our decision point. Our realization, or our imagination, of what fills out the bounded horizons of possibilities for the future is constrained by an awareness of the economic institutions that delimit our actions and color their outcomes. In such conditions our decisions are made. But in economic affairs we can never be sure of where we are going to be, or where, if we take certain actions, we shall arrive as a result. We can never know the end or the conclusion to which action will actually take us. We know only where we stand in the unique decision moments in the flux of time. Our task is that of interpreting our history, our present, and our tentative future in such a way as to make, in each unique situation, our best next move. How, then, can our decision moments have decisive meaning, and how are our decisions themselves determined? Historic time enters economic analysis when the passing of it influences our grasp of the uniqueness of the decision moments at which we stand. It casts its kaleidoscopic light on the imaginative perception of possibilities for the future that offers us the skein of conceivable outcomes and actions among which we choose. In the following sections of this chapter we shall allow these perceptions to throw further light on the theory of the firm and its employment of money capital. Possibility and economic decisions To begin an answer to these questions, we confront an issue that has partially intruded into our previous discussions. To what extent, we have asked, can the variables that influence decisions be properly understood as distributional variables, or subject, that is, to description by assigned probability distributions? This question raises acutely the meaning and treatment of uncertainty. The matter at issue is not simply that of uncertainty as distinct from risk, though in the fashion of Knight, Keynes, Shackle, and others, we have highlighted that distinction. The issue is grounded in the fact that the human perception of uncertainty is bound up with the perception of the passing and significance of time. Economic decisions are always decisions in time, temporally bounded and having temporal referents. The complexities of time cannot be elided by the conception that our vision of tomorrow can be safely delimited by our knowledge of yesterday. Let us imagine that a statement of probability is before us. This may be a statement that in a large number of conceivable experiments there is a wellattested reason to believe that in a specified proportion of the total number of outcomes a designatable result will emerge. In this sense, statistical probabil-

214 III Postclassical perspectives ities have meaning and applicability in economic affairs, in life insurance mortality tables, for example. In general, the experiments to whose outcomes probabilities are attached must be seriable, or replicable, and no aspect of uniqueness should surround the experiment to prohibit its recall and subsequent reproduction. On the other hand, the statement of probability may purport to be a description of the likelihood that in a single experiment, or as a result of a single and unique decision, a designated possible result will emerge. In that sense, the statement of probability has no meaning. The assignment of probabilities to future possible outcomes, either in the form of assumedly objective probability distributions or of subjectively assigned probability distributions, is actually an assumption of knowledge. It is this that makes the entire probability machinery inapplicable to decisions under uncertainty. For knowledge is the antithesis of uncertainty. It is the abolition of uncertainty (see Shackle, 1969, Ch. VII). The assignment of a probability distribution to possible outcomes involves the assertion that if the same event were to be repeated a large number of times, a designatable outcome would emerge in a prespecifiable proportion of instances. But such a replicability of experiments is precluded by the unique character of most economic decision situations. The taking of economic decisions frequently destroys forever the possibility of their being taken again. Decisions are in this sense "self-destructive." 1 For probabilities to be assigned over a set of possible outcomes, the assumption must be made that the set is complete and exhaustively descriptive of the possibilities without a residual hypothesis. As a homely example due to Shackle, imagine that of six candidates for a job one is, on all the grounds we can contemplate, the most qualified for the position. We accordingly assign probabilities of success to each of the six candidates with this imagination in mind. The probabilities assigned sum to unity and we have, supposedly, a bona fide probability distribution of the outcome. Suppose now that a seventh candidate is introduced, and that he appears to possess much the same qualifications as five of the six previous candidates. Now when we introduce the seventh individual to the candidate set, we assign a probability distribution over the seven possible outcomes. But we still have the necessary summation of probabilities to unity only by taking away part of the probabilities already assigned to one or more of the other six. If we try to capture the uncertainty The arguments in this and the following sections are heavily dependent on the work of Shackle, who has developed, in numerous books and articles over the last three decades, a critique of probability theorizing and has constructed an alternative potential surprise decision apparatus. These have been subject to extensive evaluation and criticism. See Shackle (1969, Ch. XI) for Shackle's evaluation of some of his critics, the recent extensive critique by Ford (1983), and Vickers (1978, Ch. 8). See also footnote 2.

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by treating the outcome variable in this way as a distributional variable, we are making the assignment of probabilities dependent on the number of possible outcomes in view even though the very meaning of probability assignment implies the assumption that the distribution encompasses an exhaustive specification of the set of possible outcomes. We have supposed that the seventh individual possessed qualifications that were comparable with those of the other five and were again quite below those of our previous outstanding candidate. We therefore ask the telling question: How surprised would we now be if the outstanding candidate were awarded the job? Would we be less or more surprised than we would have been in the original situation? The introduction of the seventh candidate may make no difference at all to our contemplation of the outcome so far as the outstanding candidate is concerned. In the light of this, surely a means of handling uncertainty in decision-making situations needs to be devised that can take account of precisely this kind of possibility. This need arises because of what we have identified previously as the uniqueness of decision situations. Let us ask by way of further illustration whether it would have been meaningful for Napoleon to have assigned a probability distribution to the outcome of the battle of Waterloo. We may answer in the negative on the grounds that if he won the battle, there would be no need to repeat it, and if he lost the battle, it would not be possible to fight it again. In many economic situations, in investment decisions in the firm, for example, the decision events are unique in the same sense as was Napoleon's situation. The taking of the decision forever changes the decision environment and the decision nexus beyond recall. For these reasons, it appears necessary to replace the distributional analysis of probability by an analysis that focuses on the possibilities of outcome magnitudes of nondistributional variables. Potential-surprise function Let us approach the decision-making problem from the opposite side from that on which probability thought forms are imagined to be applicable. By asking the decision maker to describe the probabilities he would assign to certain specifiable outcomes, we are asking him to base his judgment on degrees of belief in conceivable results. Let us instead envisage a procedure that attaches degrees of disbelief to possible outcomes. We may imagine that when confronted with a range of possible outcomes, the decision maker specifies, on an ordinal scale, the degree of potential surprise he feels now, at the decision moment, he would experience in the future if various outcomes were to occur. By focusing on degrees of disbelief in the description of possible outcomes, our argument will be protected against any attempt to transform it

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B + AX

Figure 12.1

into an argument in probability. The logic of the following argument is that the potential-surprise function is in no sense transmutable into a probability density function (see Katzner, 1986). Of course, the moment at which, in the future, the event that was previously in view actually occurs is different from the moment at which the decision is made on the basis of the contemplated potential surprise. No meaning can attach to the question of what surprise the decision maker would get in the future if a specified event occurred. He is concerned with the contemplation now of the potential surprise with which he contemplates the various possible outcomes of his decision. To give definiteness to these ideas, let us consider the relationship described in Figure 12.1. The horizontal axis measures the values of an outcome variable X, which we can assume for the present to be money values, thus providing a continuous scale. In different applications they may be, for example, the money values of cash flows generated by an investment activity or the rates of return on assets. Outcome values to the right of the zero point in Figure 12.1 are understood to be positive, and those to the left will be negative. The vertical axis of Figure 12.1 measures, on an ordinal scale defined on an arbitrary choice of origin and unit of measurement, the degree of potential surprise that the decision maker attaches to the various possible outcomes. To the possible outcome of OC, for example, is attached a potential surprise of CD. That magnitude will have been arrived at in answer to a question regarding the degree of disbelief with which the possibility of an outcome equal to

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OC was envisaged, or the degree to which, in his present state and manner of seeing things, the individual imagines he would be surprised to see the value OC emerge. Proceeding in a similar fashion, it is possible to build up the entire curve y = y(x) describing the potential surprise attached to the entire range of possible outcomes (see Vickers, 1978). In contemplating the possible outcomes from a decision event, an individual will conceivably be able to specify a positive magnitude so large that he entertains an extremely high degree of disbelief in its possible occurrence. This may be so high that he does not regard the outcome to be possible at all. He regards it as perfectly /^possible. A maximum degree of potential surprise is thus attached to that outcome magnitude. We have labeled that maximum potential surprise as y in Figure 12.1. Drawing a horizontal line at that ordinate, we have shown the potential-surprise curve as asymptotic to it in the easterly direction. Similarly, the loss outcome segment of the potentialsurprise curve would be asymptotic to the maximum potential-surprise magnitude in the westerly direction. The rates at which the respective positive and negative segments of the potential-surprise curve approach the asymptote will not necessarily be the same. The one side is not necessarily the mirror image of the other. Everything depends on the determinants of the decision maker's imaginative view of things at the unique decision moment at which the various outcome possibilities are contemplated. Moreover, the potential-surprise curve should perhaps in certain instances be truncated in the loss direction. Suppose, for example, that the X values represented wealth outcomes from an investment activity and that the magnitude M on the horizontal axis represented the individual's total wealth and therefore his maximum sustainable loss. In that event, the loss segment of the curve to the left of the point N, which is vertically above the maximum sustainable loss M, has no meaning. For decisionmaking purposes, the curve should be truncated at that point. Consider now the segment AB on the X variable axis. Ignoring for the moment the remainder of the curve, imagine that in the decision event being contemplated the entire range of possible outcomes is described by a potential-surprise curve coinciding with the X axis throughout its length and terminating in A and B. It can then be said that the decision maker is attaching a zero potential surprise to all possible outcomes in this range. As he sees things, he would not be at all surprised if any outcome in the range AB occurred. For purposes of vocabulary, all outcomes in the range AB are conceived to be perfectly possible and have associated with them zero potential surprise. Continuing to imagine a function that terminates at A and B, the individual will, we suggest, focus his attention on only the best and the worst possible outcomes the project offers. In reacting to the ability of this project or eco-

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nomic opportunity to arrest or grasp or command his attention, the individual will find his attention focused on the magnitudes A and B. These will be referred to as the focus elements, the focus loss and the focus gain of the project. For there is no point in an individual's being interested in the fact that an activity may bring a moderate fortune, say within the range of OB, if it is equally possible, or in this case perfectly possible, that it will bring a considerable fortune, OB. At the same time, there is no point in being concerned that the activity may result in moderate misfortune, say within the range of OA, if it is likewise perfectly possible that it will result in considerable misfortune, OA. "When many different things are all equally and perfectly possible," Shackle observes, "it is the brilliant and the black extremes which hold our thoughts" (1969, p. 118; see also Shackle, 1970, Ch. 5). In the case in hand, then, we can characterize the attention-arresting power of the project in terms of its focus elements A and B. In more complex and realistic cases, it is perfectly conceivable that within such a range "zero potential surprise can be assigned to each of an unlimited number of rival, mutually exclusive [outcome] hypotheses all at once; any number of suggested answers to a question . . . can all be regarded . . . as perfectly possible. The same is true of any other degree of potential surprise. . . . Potential surprise is completely nondistributional" (Shackle, 1969, p. 70). Suppose now that a project being contemplated differs from the one we have just considered in that its range of possible outcomes extends along the X axis to the right of B and also possibly to the left of A. Its potential-surprise curve can still be as shown in Figure 12.1, having an inner range of perfectly possible or zero potential surprise outcomes, but with the prospect also of other possible outcomes to which positive potential surprise is attached. Examine now the rightward segment of such a project's potential-surprise curve. A slightly higher outcome than B, say B + AX, may be conceived to be possible, but less than perfectly possible, and it may have associated with it a positive potential surprise indicated by the point W on the curve. As the level of possible outcome is higher, the degree of potential surprise will also become larger. Indeed, in view of the increasing degree of uncertainty surrounding the higher possible outcomes, and considering the fact that the possible outcomes become more shadowy in the decision maker's vision, the potential surprise may well increase for a time at an increasing rate. But for very high possible outcomes, for those, for example, in the range that is thought to be nearly impossible, the rate of increase in potential surprise will have slackened, and the potential-surprise curve will approach the asymptote y. It follows that the rightward segment of the potential-surprise curve is likely to assume the elongated S form shown in Figure 12.1. A comparable argument

12 Uncertainty and decisions in the firm

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can be applied to the S form of the leftward or loss segment of the potentialsurprise curve also. In the first of the projects we discussed, the project's power to arrest and command the decision maker's attention was encapsuled in the gain and loss focus values B and A. For the second project, as the contemplated possible outcome increases, say to B + AX, the attractiveness of this value, or its power to grasp or focus the decision maker's attention, may outweigh the nonzero degree of potential surprise associated with it. Thus the focus element of the project may shift from B to W. Then whereas the first project commanded attention by virtue of its focus elements A and B, the second project commands attention by virtue of its focus elements A and W. We can visualize in this way a cognitive and evaluative process in the mind of the decision maker whereby he marshalls together, with respect to a given project, both (i) the range of possible outcome elements associated with the project and (ii) the degree of uncertainty associated with each such element or subset of the total range of elements. If we assume that the decision maker is risk averse, to employ the parallel thought form we encountered in the theory of utility in Chapter 6, then he will be (1) attracted in a positive sense by higher possible money values of possible outcomes but (2) repelled by the higher degrees of potential surprise associated with them. A trade-off evolves in the decision maker's mind. He will want to choose projects or commit economic resources in such a way that if he is risk averse, he will have the prospect of realizing as high as possible a value outcome without the potential-surprise curve at such a point having risen too far above the X axis. Investment decision criterion We now recall that each potential-surprise curve that the decision maker confronts will refer to a specific investment project or other economic opportunity. If investment projects are in view, the decision maker will confront a nest or series of such potential-surprise curves, one for each project available for consideration. In respect to each potential-surprise curve, the decision maker must confront all possible outcomes extending over the range over which the curve is defined, but he must regard these not as possible occurrences in a series of repetitive processes but as rival outcomes. Only one of them can eventuate, determined by the unique conjunction of forces that give rise to it. Not only are all of the possible outcomes rivals in this sense, but the decision situation in which the decision maker stands is unique in the sense in which we have explained its uniqueness before. It is unique in that it occurs at a unique point in historic time, in the fact that the decision environment, existentially and

220

III Postclassical perspectives

epistemologically, is therefore unique, and unique in the fact that the taking of the decision may destroy forever the possibility of its being taken, or even contemplated, again. But we can deduce a still firmer statement. As all of the possible outcomes of a given project over which its potential surprise function is defined are rival outcomes and only one of them can occur, no meaning at all could be attached to any kind of averaging of them, for no repetitive processes are available to give any rationale to such an averaging process. Potential surprise is completely nondistributional. No logical meaning can possibly attach to any such concept as the expected value or the variance of possible outcomes in such unique situations. Consider now a potential-surprise function of the kind described in Figure 12.1 associated with a specific investment project A. We now describe on the outcome axis the possible net present values of the project. We concentrate separately on its positive, or gain, and its negative, or loss, segments. We wish to isolate, in a manner consistent with the basic development to this point, what we called focus elements, or magnitudes on the value outcome axis on which the decision maker can concentrate his attention for purposes of evaluating the project. On the gain side, for example, such an element may be the value magnitude associated with points W or D on the potential-surprise curve in Figure 12.1. The focus element summarizes, as we observed in our development of the underlying potential-surprise function, the power of the investment project to arrest or command the decision maker's attention. Such an attention-arresting function, designated R in Equation (12.1), or what has been previously referred to as an attractiveness function (Vickers, 1978, p. 148), will be defined over the possible-outcome and potential-surprise magnitudes, x and y, respectively: R = R(x,y)

(12.1)

In Figure 12.2 are shown the isoattractiveness contours implicit in the attractiveness function described in Equation (12.1), together with a potentialsurprise curve. As will be indicated immediately, that focus element will exert the maximum attention-arresting power, or will provide the maximum attainable value of the attractiveness function, which establishes an equality between the slope of the isoattractiveness contour and the potential-surprise curve. The arrows in Figure 12.2 indicate the directions of increasing attracting power of possible focus points. Given the relation between the potential-surprise and outcome values, v = y(x), and recognizing that the potential-surprise function operates as a constraint on the possible values of the attractiveness function, or that the operative values of the latter function will be determined by coordinates that lie on the potential-surprise function, the attractiveness function becomes

12 Uncertainty and decisions in the firm

221

Figure 12.2

R = R[x, y(x)]

(12.2)

Taking R\ and R2 as the partial derivatives of the attractiveness function with respect to its first and second arguments, respectively, the attention-arresting power of a project is summarized in the following relations: R2 < 0

for 0 < y < y, x # 0

(12.3)

In determining the gain and loss focus elements, the decision maker will investigate the (JC, v) coordinates that generate the maximum value of the Rfunction in each of the gain and loss directions, understanding, as has been noted, that the relevant coordinates will lie on the potential-surprise function. The decision maker's subjective reaction.to possible outcomes and potential surprise summarized in Equations (12.1) and (12.3) therefore implies that the focus elements of the investment project under examination will be the points at which the subjective outcome-surprise trade-off implicit in Equation (12.1), dyldx = -RXIR2

(12.4)

is equal to the corresponding trade-off defined in the potential-surprise function that is relevant to the project. That is, dx

PS

dx

(12.5)

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III Postclassical perspectives

where the left-hand and right-hand sides of Equation (12.5) represent, respectively, the slope of the potential surprise function and the slope relation between the arguments of the attractiveness function. Such points occur at T and V in Figure 12.2, reflecting focus values of R and S on the x axis immediately below them. At those points, the highest attainable values of the /^-function will be registered. No possible (x, y) point on the potential-surprise function would generate a higher power of attraction or attention-arresting power as summarized in the attractiveness function. The point of maximum attainable R-value, therefore, will describe the (x, y) coordinates on the potentialsurprise function that serves as the focus element. Corresponding arguments thus lead to the specification of focus elements on both the loss, or negative, and the gain, or positive, segments of the potential-surprise function. The focus values describe the projects that the decision maker will subject to particularly careful scrutiny in the course of making his investment decision. He will be interested in those projects that promise high favorable outcomes with low potential surprise and/or large possible losses or unfavorable outcomes with low potential surprise. In his final project selection, the decision maker will be positively disposed to the high possibility, or low potential surprise, of large gains. Similarly, he will prefer not to invest in projects that hold out the prospect, with high possibility or low potential surprise, of large losses. In the example of the investment project, the loss could be visualized as negative net present values. We may therefore make use of the focus elements to establish an investment decision criterion in the following manner. In relation to the investment project, we now possess four pieces of information or input data for the decision event. These are (i) the focus element on the gain side, which we shall refer to in the following argument as FG, or, in the case of project A, as FGA; (ii) a comparable focus element on the loss side, FLA; (iii) a potential-surprise magnitude associated with the focus gain value, PS G A; and (iv) a potential surprise associated with the focus loss value, PS LA . The potential-surprise curve for project A with which we are working may not extend into, or may not extend significantly far into, the loss segment. Economic conditions may be generally so favorable that the firm can expect, with a high degree of confidence, to realize a positive outcome, or at the worst virtually break even, on the project. In such a case, the loss segment of the potential-surprise curve may be virtually vertical at or near the zero point on the outcome axis. What we are here referring to as the loss region may therefore include small positive values. The final investment decision leading to an acceptance or rejection of project A involves a subjective weighting in the mind of the decision maker of these four pieces of information. We may therefore conceptualize a Decision Index (DI) and stylize it as

12 Uncertainty and decisions in the firm DIA = /(FG A , PS GA , FLA, PSLA)

223 (12.6)

When the decision index value relevant to the project has been determined, the investment criterion can be visualized as DI ^ 6

(12.7)

where 6 describes a critical required level set by the firm. The project will be acceptable to the firm only if it has associated with it a Decision Index value no smaller than the prespecified critical level. This critical value may, of course, be zero. It will be set by the decision maker at a level that reflects his general aversion to uncertainty, as well as, in appropriate cases, the need to concentrate resources on investment projects that promise a rapid return cash inflow, or a low payback period. This may be desirable in situations, such as we encountered in the preceding chapter, where cash flows are highly desirable as a means of financing further investment outlays that facilitate the continued growth of the firm and promise highly attractive benefits in the longer run future. The general form of the Decision Index function in Equation (12.6) can be further specified as follows. Writing/, / = 1, . . ., 4, to refer to the partial derivatives, indicating the directions of change of the function value with respect to the four arguments in the order in which they are stated in Equation (12.6), we have /i>0,/ 2